What does black do after Be2? I'm not understanding.
the english muffin

I do not know about calling it the English Muffin, how about the Dunder Muffin?
1.e4 e5 2.Nf3 Nc6 3.Be2 Bc5 4.Nxe5 Nxe5 5.d4 Bxd4 6.Qxd4 Qf6 seems best in your variation. Black is a little awkward, but black threatens discovered check so he should be fine here.
The move I want to play is 4...Qh4
1.e4 e5 2.Nf3 Nc6 3.Be2 Bc5 4.Nxe5 Qh4 5.g3 Qxe4 6.Nf3 Nd4! and black wins
The best line for white is
1,e4 e5 2.Nf3 Nc6 3.Be2 Bc5 4.Nxe5 Qh4 5.Nd3 to keep the pawn
5...Bb6 6.0-0 Qxe4 7.Re1 Nge7 and black will castle and will be safe while white's pieces are a little misplaced.
Also possible is 5...Bb6 6.Nc3 Nf6 when 7.e5 is met by 7...Nxe5!

Position is equal. A much better and more common variation of this opening as black occurs in the two knights defense.

The fork trick is a great equalizing resource for Black, but I see little advantage for White to do it. White wants to maintain his initiative not equalize.

Blackburne was the first person who have played 3.Be2. see the game here: http://www.chessgames.com/perl/chessgame?gid=1029074
The most common response to 3.Be2 is Nf6
I don't think Bc5 is the best response to Be2 since it would allow white to do the fork trick.

In the spirit of using the central fork trick, how about
1 c4 e5 2 Nc3 Bc5 3 Nf3 Nc6 4 Nxe5 Nxe5 5 d4
as something that could be called the English Muffin?

greyboy, 1 c4 e5 2 Nc3 Bc5 3 Nf3 Nc6 4 Nxe5 Nxe5 5 d4 and where you give 5...Bd6 white should play 6 c5.
Black needs to play either 5...Nxc4 or 5...Bxd4.
I still say my example of the English Muffin using the central fork trick from post 13 is good.
This is an opening I made; The English Muffin.
1.e4, e5. 2.Nf3, Nc6 3.Be2; this move is critical because if you do Bc4, the trick won't work, and you would get behind a bishop; Bc5; if he doesn't do this, the opening is dead; 4.Nxe5; see where this is going?; Nxe5. 5.d4!
that is the regular English Muffin. If extended and you get lucky...Bxd4. 6. Qxd4, Ng4. 7.Qxg7, Qf6. 8. Qxg4! and from there, play you're game!
good luck :)