Queens gambit accepted - 3.e3 or 3.e4?

I've found that regarding getting the pawn back, if you play 3. Nf3 they can steer the game into a Noteboom Slav which white has to understand how to play against. I don't think 3. Nf3 is so good for fast games like 5 mins blitz, since it's often centred around positional maoeuvring on the Q-side.

Here are three answers and they are not easy.

The first is based on a computer that already beats grandmasters so its rating should be better. The latest Stockfish version prefers 3.e4 but the one before that 3.Nf3

The second answer is based on analyzing the games of players of your level according to what they play. For example, for players rated 1800-2000, perhaps 3.e3 is best because there is the highest success rate for White. But here it requires a longer analysis and calculation. For example, after 3. e4 the best answer for Black is 3. ...e5 (or even 3. ...Nf6). But 1800-2000 level players most often play a very weak 3...e6 and White has a good game. So it is necessary to calculate what my opponents play most often and choose the opening accordingly.

The third option is to find your favorite player and see what he plays. For example, Caruana plays 3.Nf3

I think they are all about equally strong, with different plusses and minuses. I've played them all, and also faced them all as black since I have often played the QGA myself. I wouldn't have an issue with any of those three options as a part of a student's repertoire.

If I had to choose one I like the most, probably 3. e4 but that's just a personal choice. The difference between these three moves is pretty subjective, all roads lead to like +0.1 or +0.2 and ultimately black is fine if they have done their homework.

I agree that 3. Nc3 makes black's job a lot easier, and 3...a6 is the typical recommendation although there are other ways.