You could certainly Dash first to increase you movement allowance, then attack the first mook, then move, then attack the second mook. That's not exactly what you're asking about though, since you want to see the result of the first attack before deciding to Dash.
In the strict rules as written, however, only movement is explicitly allowed to be broken up arbitrarily. Taking the Attack action doesn't "add to your attack allowance" the way Dash adds to your movement allowance.
Implicitly, when you take an action, you are committed to completing the entire action unless you have a rule that says otherwise. "Moving Between Attacks" is one such rule, and object interaction (PH p.190) is another:
You can also interact with one object or feature of the environment for free, during either your move or your action.
By this reading, you have to complete all the Extra Attacks that are part of the initial Attack action before using Action Surge to Dash.
You can't take both Dash and Ready actions
Normally you don't "prepare a reaction". You take the Ready action instead. See Player's Handbook, page 193, "Actions in Combat"
Sometimes you want to get the jump on a foe or wait for
a particular circumstance before you act. To do so, you
can take the Ready action on your turn so that you can
act later in the round using your reaction.
Then you use you reaction (if still available) to actually perform the readied action when trigger occurs. You spend both your Action and your Reaction in this case. Therefore, you can't use your action to Dash after you declared a readied action, because it is already spent.
Unless you have an extra Action, or you're a Rogue
As notices by @NeilSlater, a Rogue has his Cunning Action feature, which allows him to use a bonus action for Dash. You can take the Ready action afterwards. However, this doesn't give you more movement than usual, since you just can Dash twice.
Also, you can't use "my turn ends" trigger
PH requires "perceivable circumstance" for a trigger:
First, you decide what perceivable circumstance will trigger your reaction.
Turns are game mechanics, they don't exist in the game world, hence, are not perceivable by the character, see Can you ready an action for "immediately before my next turn"?
You can move up to your speed using the Ready action
However, the question remains — what if you spent all your movement, and then take the Ready action to move 30ft more? Yes you can do this, providing your speed is 30 (or more) feet:
you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it
Ready action works like a "delayed Dash" in this case.
Best Answer
A character can use Dash multiple times, given the actions to feed them. The Speed limit doesn't prevent multiple Dash actions and using their granted movement, as exceeding your Speed in a defined way is explicitly what Dash does.
That's not why the Amazing Lightspeed Horse doesn't work. There's no conflict between those two answers.
The reason the Amazing Lightspeed Horse doesn't work is because it's not using multiple Dash actions. It can be ordered to Dash by as many riders as one might like, but it still can't execute such an order when it has run out of actions (bonus or otherwise) with which to Dash.
Pre-empting the counter-arguments
“But a mount uses its rider's actions.”
A mount doesn't use its rider's actions. A controlled mount uses its own actions to execute the rider's directions (PHB p. 198, emphasis mine):
So the poor horse getting an infinite number of riders is still limited by its own allotment of actions no matter what blizzard of orders it is bombarded with.
“A mount gets a new turn and new actions for each new rider.”
It's true that a mount gets new actions on a new turn, but the mount isn't getting a new turn with each new rider in a round. Controlling a mount only makes the mount to use its (singular) turn for the round synchronously with the rider's turn. The following is the entirety of the rules creating this concept of overlapping turns, and it can't be read as creating a fresh turn for the mount (ibid):
Swapping infinite riders doesn't give the horse infinite fresh turns and the fresh actions that would in theory come with those turns.
Thief's Reflexes ability
Yeah, this works. A 17th-level thief who gets two turns in the first round of combat gets all the actions they can execute. There are two things to consider:
Can you move your Speed on each turn, or just once because it's one round?
You can move your Speed on each of the two turns, as defined by the core Movement and Position rules (PHB, p. 190):
Can you Dash multiple times?
As many actions as you have to devote to a Dash action each turn can be used to Dash. For a Thief with Thief's Reflexes and Cunning Action, that means a total of 4 actions (2× Action, 2× Bonus Action) could be devoted to increasing available movement over the course of the two turns.
As a result, two turns in one round makes for being able to move Speed feet twice and adding Speed feet to available movement four times, for a total available movement over those two turns of 6× one's Speed.