How would you handle it?  I know stress goes away, but how would you handle "healing" minor or moderate consequences magically?

For those two categories of consequences, it doesn't seem like anything extra is needed.

Straight from the book:

Mild consequences are Bruised Hand, Nasty Shiner, Winded, Flustered, Distracted.

Moderate consequences are Belly Slash, Bad First Degree Burn, Twisted Ankle, Exhausted, Drunk.

Severe consequences are Broken Leg, Bad Second-Degree Burn, Crippling Shame, Trauma-Induced Phobia.

Recovery for mild consequences is started by first aid (a Scholarship roll) and they go away at the end of the next scene.

Recovery for moderate and severe consequences is started by treatment by a doctor. Moderate consequences recover after the next session. Severe consequences last until the end of the next scenario.

For magical healing, the Reiki Healing spell (p. 300) makes a moderate consequence go away like a mild consequence. With this spell severe consequences can optionally be made to recover like moderate ones, for an increase in power (essentially casting time or preparation).

That means you can recover from any mild or moderate consequence by the end of the next scene. Given that most fight scenes are followed by scenes of aftermath and travel to the next action scene, you should have everything you need with the basic rules.

Harry has(had) Wizard's Constitution.  He doesn't need 'an excuse to start the healing process' - he never actually needs to visit a hospital - and, yet, Butters patches him up.  Obviously because it has some kind of effect.  Is it purely narrative?  Is it just to break up the pace of the novel?  Give him a chance to recover his stress track?  Catch his breath?  It could be but I don't like that.  If I was playing Butters as a character,  I would want all those procedures to actually help Harry's character otherwise I've wasted a stunt.

You're trying to reduce the novels to numerical computations, whereas I think the game designers' intent is to make our games play like the novels.

There is no game mechanical reason for Butters to do what he does, because the novel is not a game. Butters patches up Harry because Harry needs to relate to regular people. We need to see that Harry is human. This becomes more obvious later in the series when Harry begins to seriously fear that he's losing his humanity to the powers that threaten to take over his soul.

The medical treatment is also about character development, and Butters can give Harry some insight into wizardly biology that Harry can't get by himself. The treatment defines the relationship between Harry and Butters, draws Butters into the web of complexity that's Harry's life.

The novels are about these people, not the mechanics of the battles. In later novels Butters' investment in Harry becomes a pivotal element in his life and the others around him. If Harry never visited Butters for treatment a major plot point would never happen, because Butters would be completely useless to Harry. Butters, being mundane, is also responsible for keeping Harry grounded in the mortal world and not getting completely lost in the faerie world.

When we play these games it is tempting to reduce human beings to mere stats and power lists. But novels that read like the transcription of a computer game are not interesting. I find the most successful play sessions are those that unfold without us ever having to roll dice or consult the rule book -- when the action unfolds organically out of the tensions between characters and the story that the GM has created for them to interact with.

The Fate system is set up to give the players a lot of control over the direction of the scenario. Players can spend fate points and make declarations to make things go the way they want to. The mechanics can be manipulated so that players themselves can usually decide when they're going to succeed or fail at something.

If making your players sit around waiting to heal is a waste of time in your game -- that is, your characters have no need to interact with a doctor character like Butters -- then just toss the rule.

You use the phrase "narrative reasons" like that's a bad thing. In my mind, the narrative -- the story -- is the only thing. The stats and the dice rolling are there to spice things up, add some randomness and take some of the arbitrariness that can develop if GMs can just dictate every outcome.

As I said in my first post, many people get around the 'I need an excuse for healing' by taking wizards constitution and, in fact, it's a better use of your refresh than taking the doctor stunt unless you are pure mortal.

Taking the Doctor stunt allows you to start the healing process for moderate and severe (in a hospital) consequences afflicted on any number of mortals. I assume that most GMs won't allow someone with Doctor to treat themselves for anything but mild consequences, so you'd never take it to treat yourself.

Inhuman Recovery and Wizard's Constitution only work on the character who takes the power, not on anyone else. That's the real difference here. If your campaign has no pure PC mortals who need medical treatment, then yes, the Doctor stunt is useless. But if the PCs in your campaign are mostly Pure Mortals, then it means that one character can "heal" all the others. If that character took Inhuman Recovery or Wizard's Con, he couldn't help the others heal in any useful way.

Remember, Pure Mortals get two free refresh for the inconvenience of not having access to special powers. Is it "worth" it to play a character like Murphy, who's an expert at a lot of mundane things, instead of a powerful wizard like Harry or a White Court Vampire like Thomas?

What we're talking about here is character concept. If your concept is a Pure Mortal, you don't have the option of taking those powers. All it really means is that players of Pure Mortal characters will have to depend on NPC doctors if no else wants to spend refresh on Doctor.

I view the city planning phase as the players making a statement about the kinds of scenarios they want to play, the kind of world they want to experience; essentially, the kind of book they want to read. The character creation phase then figures into that: most of the players had origins and first adventures that derive from aspects built into the city.

I remember reading advice about scenario creation that taking the aspects of player characters, NPCs and the city, and using them to create the bones of scenarios. By choosing aspects and character templates, characters are making statements about what their goals are, and what kinds of scenarios they want to be involved in.

The game is still built on the GM/player or playwright/actor paradigm: one person is responsible for writing a play in which the characters will improvise their roles.

I've played games (like Amber) in which players actively pursued their own goals and made things happen without regard to the plans of the GM. We had a campaign with six or seven players, and alternated between three GMs. In almost every case, the players who instigated their own plots were the other GMs. A lot of players just aren't comfortable with forcing their agendas on other players.

The other factor is idle time during the session: if players go completely off on their own tangents they run the risk of leaving the other players sitting there twiddling their thumbs. Getting all the players involved in the action can sometimes be hard, especially when you've got six players.

When I started our DF campaign, I really wanted to get the players involved in city creation and plotting so that I could let players take over as GM for specific scenarios that they want to see happen. We've been playing for a few months now, and so far no one has said they wanted to do it. Running a game is a lot of work.

I think the Fate system has all the tools to allow maximum player contribution to the game, but the reality is that players either don't want to step on other people's toes by forwarding their own agendas, or can't spend the time and effort required to initiate their own storylines.

That means, for the most part, players rely on the GM to integrate all the players' goals into a coherent campaign that will let everyone have the kind of experience they're looking for.