I remember a GM comment about d3 Bc5 to the effect that he did not understand all the nuances, but decided that no one else did either. The same may apply to  to d3 Be7. There are annotated games in ChessPublishing. When it comes to a recommended repertoire for black there is Lokander's Opening Repertoire, previously mentioned. That's all I'm aware of. One of his lines goes

1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. d3 Be7 5. O-O O-O 6. Nbd2 d6 7. c3 Na5 8.
Bb5 a6 9. Ba4 c5 10. Re1 b5 11. Bc2. My note is that this has transposed to the Pilnik system of the Ruy Lopez:
1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O! Be7 6. Re1 b5 7. Bb3 d6 8. c3
O-O 9. d3 Na5 10. Bc2 c5 11. Nbd2.
I find it remarkable that two very different move orders arrive at the exact same position on tempo. I'm not aware of a way for either side to steal a tempo.

I could not find a forum entry on 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. d3 Be7. It's a main line, second only to 4...Bc5 in popularity. 4...Be7 was recommended by Martin Lokander:

https://forwardchess.com/product/opening-repertoire:-the-open-games-with-black?s...

The play can continue 5. O-O O-O 6. Re1 d6 7. a4. At this point Lokander prefers 7...Kh8 to the more frequently played 7...Na5.

I am curious about treating this like a Chigorin Ruy Lopez. In my notes I have 7...Na5 8. Ba2 c5 9. Na3 Nc6 10. c3 Be6 11. Be6 fe6 "slightly more pleasant for White." Continuing on a little, 12. Bd2 a6 13. h3, black seems to have good chances in an unbalanced game (13..h3, 13...Qe8, 13...Rc8).