It's option 2: Both sides roll, then decide to start invoking aspects.
This comes across in How To Do Stuff: Outcomes, Actions and Approaches. First, you choose your action and approach, and then:
Roll the Dice, Add Your Bonus
Time to take up dice and roll. Take the bonus associated with the approach you’ve chosen and add it to the result on the dice. If you have a stunt that applies, add that too. That’s your total. Compare it to what your opponent (usually the GM) has.
(emphasis added — both you and your opponent have now rolled, and the results are all out in the open...)
Decide Whether to Modify the Roll
Finally, decide whether you want to alter your roll by invoking aspects—this is discussed a lot in Aspects and Fate Points.
Basically, in Fate, the score you're opposing is always known before the point you start making invocations. Always! Otherwise you have no idea how many invocations you might have to bust out. Hiding difficulties in Fate has a negative impact, elaborated in this question about Dresden Files.
The fundamental concept of this is that invokes should never have to be made speculatively; Fate points are the currency which defines the ebb and flow of the game experience, and as such are precious commodities which need to be spent meaningfully.
Yes, this can result in a back-and-forth invocation competition until someone pulls out, but that's okay in practice.
So who gets to invoke first?
Fate gameplay is very much a conversation. There's no "last in, first out" or "I called it first" in the mechanics of invoking; instead, the players have space to discuss their choices and everyone gets to make equally informed decisions (whether the characters have the info or not).
I've seen "invocation escalation" resolved with both players realising they're willing to go all out, and agreeing not to play any invokes at all because the outcome would be the same except they won't have cool things left for later on. I think most groups would be okay with that, and in that context who got to start the invokes doesn't matter at all.