I'm a "catch the dive boat at the Caleta at 0920" sort of person, and I can't even prove anything actually exists on the island between 0200 and 0800. I'm honestly not sure how big the demand is for tourist-oriented breakfast before even the mosquitos wake up. I have been told by reliable informants (but will likely never confirm in person) that there are lots of places to get a good breakfast early in the morning, but some such places cater more to working islanders who are more interested in quick service and low price, and who often want a taco, club sandwich, or torta for breakfast.
I know from the signs that Restaurant Tamarindos opens at 0600. They have amazing lechon tacos or tortas on Saturday and Wednesday and cochinita pibil other days, relleno negro, etc. You can get tacos, salbutes, tortas as well as eggs. I just now learned from their website that they have papadzules! They're on 30 Av at the corner of Calle 7, across from Meridiano 87. They have only a few tables and do constant take-out business. This is where I get my final breakfast when I'm leaving the island, along with tortas to take on the plane. I could probably auction those things off as soon as the other passengers become aware of them.