Stack tortillas, seal in foil, and heat in a 375° oven until hot, 10 minutes.
In a 12-inch nonstick frying pan over medium heat, melt butter. Break eggs into pan, slightly apart; cover and cook just until whites are almost firm and yolks are still soft, about 3 minutes.
Meanwhile, spread 1/2 cup lettuce on each plate. Lay tortillas on lettuce and spoon beans over tortillas. With a wide spatula, transfer an egg to each serving. Spoon tomato salsa around one side of eggs, green around the other side. Sprinkle with cheddar cheese, then cotija.
Garnish as desired with avocado, jalapeño chilies, sour cream, and lime wedges; add salt to taste. Enclose remaining tortillas in a linen napkin to keep warm, and serve alongside.
Bacon black beans: In a 10- to 12-inch frying pan over medium-high heat, stir 4 oz. chopped bacon (4 slices) until crisp and brown, 5 to 7 minutes. With a slotted spoon, transfer to paper towels to drain. Spoon all but 2 tablespoons fat from pan (save for huevos, preceding, if desired, or discard) and add 2 minced or pressed garlic cloves and 1 cup chopped onion. Stir often over high heat until onion is limp, about 3 minutes. Meanwhile, rinse and drain 2 cans (15 oz. each) black beans; add to pan along with 1 cup fat-skimmed chicken broth. Stir often until most of the liquid is evaporated, 5 to 7 minutes. Stir bacon into beans. If beans have gotten too dry, thin with a little more broth.