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.
This answer is wrong. Ignore it, unless it makes your games more fun, in which case, ignore all others.
Yes, you can, and with even more versatility than you thought.
Unlike previous editions or some other games, movement is not a type of action in D&D 5e. It is a resource that you use, almost like a currency that you spend. At the beginning of your turn, you have movement to use equal to your speed. Think of this as a deposit into your account. You can use (spend) movement at any time during your turn, in any amount you have left, any number of times. In between using (spending) movement, you can do anything that your character is capable of, including making attacks, using bonus actions, etc. If you don't have any movement left (account balance of 0), you can't move any more. After the end of your turn, any remaining movement is gone (the account is zeroed).
Using the Dash action increases the amount of movement available that turn by an amount equal to your speed. It's as if another deposit has been made. Therefore, you don't use your Dash to move, you just are able to move more because you Dashed. You can do this at the beginning of your turn, after using some of your movement, or after using all of it.
Now, how does this apply to mounts? Your mount gets a deposit equal to its speed at the beginning of your turn, and you get to direct it how, where, and when to spend that movement, as if it were your own. You can also direct the mount to use the Dash action (no action required on your part), which makes an additional deposit into its movement account. As long as the mount's movement account has movement in it, you can direct it to move as you wish, as many times as you wish, doing anything that your character is capable of in between each movement.
Note: this answer has been contradicted by a Crawford tweet since written.
Best Answer
Fighters can. Action surge means you can use one action to dash, and the other to attack. Using action surge is a feature, and does not require a bonus action.
Outside of that feature...
Bonus actions aside seems to be a weird stipulation to put on this being that that is pretty much the only way you could both dash and attack.
You move on your turn, and you are allowed to take 1 action. (PHB pg. 192). One of those actions could be Dash which gives you additional movement on your turn. Basically you are converting your action into additional movement.
However, you can break up your movement as much as you wish (PHB pg. 190), and can Action during that move (which is an attack in the case you listed). So you can definitely move AND attack, but you cannot dash and attack without using a bonus action to grant either of those things.