In a small skillet, heat 1/8 inch oil over medium to medium-high. Fry the tortillas, one at a time, until browned, about 1 minute per side. Drain on paper towels, then arrange on a baking sheet.
In a medium skillet, heat 2 tbsp. oil, two turns of the pan, over medium to medium-high. Add three-quarters of the onion, 2 cloves garlic, 1 chopped jalapeno and the cumin. Cook, stirring often, until the onions are soft, about 5 minutes. Add 3/4 cup water; simmer until the liquid is reduced to about 1/4 cup. Add the beans, half the cilantro and the hot sauce; stir until heated through, about 2 minutes. Season with salt; reduce the heat to low.
Husk, rinse and chop the tomatillos; add to a food processor. Add the remaining onion, garlic, jalapeno and cilantro, the mint, half the lime juice and 2 tbsp. water. Process until almost smooth; transfer to a bowl and season with salt.
In the processor, puree half the bean mixture and the remaining lime juice into an almost smooth paste.
Position a rack in the center of the oven; preheat the broiler.
Spread tortillas with the pureed beans; top with the whole-bean mixture, then the cheese. Broil until cheese melts. Top with the lettuce, salsa verde and jalapeno rings.