Can a Drunken Master Monk spend a ki point to use flurry of blows, but use the disengage part from the Drunken Technique feature to move to a different target before attacking with the flurry of blows?
In particular, the Drunken Technique feature states:
[…] Whenever you use Flurry of Blows, you gain the benefit of the Disengage action, and your walking speed increases by 10 feet until the end of the current turn.
I'm unsure when you have actually "used" Flurry of Blows. Is this before making either of the attacks, after making at least one, or specifically after making both of them?
No. You don't declare actions separately from actually taking them.
I address a similar point in my answer here to "Initiative: Do I lose my attack/action if my target moves or dies before my turn in combat?".
Chapter 9 of the PHB/basic rules describes how actions in combat work.
It doesn't state anywhere in the "Combat Step-by-Step" sidebar in the "The Order of Combat" section, nor anywhere in the rest of the chapter, that you declare your actions before taking your turn.
In fact, nowhere in any of the rules does it state that you must declare the action you intend to take at any point; it only says that you take them (some at any time, some when a certain condition is met or when a feature/spell lets you take them). There's nothing in the rules that requires you to declare your actions beforehand - you simply do the things you want to do on your turn, if you have the movement, action, or bonus action (or whatever other resource it takes) available to do so.
In particular, the Flurry of Blows option for the monk's Ki feature says:
This means you take the Attack action in full, then spend a ki point to make those 2 unarmed strikes as a bonus action. In particular, when you use Flurry of Blows, at least the first unarmed strike happens immediately - you don't use Flurry of Blows without at least starting to do the thing that Flurry of Blows is.
That said, the rule on moving between attacks states:
A bonus action is a type of action, and Flurry of Blows consists of more than one weapon attack (as unarmed strikes are "melee weapon attacks" despite not being made with weapons). Thus, you can move between the attacks of Flurry of Blows... But you still can't move after using Flurry of Blows but before attacking, because using Flurry of Blows entails making an attack as part of it, in the same way that using the Attack action entails making an attack as part of it.
Rules designer Jeremy Crawford unofficially confirmed this interpretation of the rules in this May 2018 tweet:
He reiterated that sentiment in a later tweet from the same month:
Crawford specifically addressed the case of Flurry of Blows in this March 2019 tweet, though he was talking about needing to actually take the Attack action first to qualify for Flurry of Blows:
Even though this tweet doesn't directly address your question, rider effects on Flurry of Blows obey a similar logic. You can't use Flurry of Blows without making one of the unarmed strikes that Flurry of Blows lets you make - there's no "declare that I'm using Flurry of Blows" step to the process. Once you start using Flurry of Blows by making your first unarmed strike as part of it, then you can move before you make the other unarmed strike that's part of it.