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 another medium skillet, heat 2 tablespoons oil, two turns of the pan, over medium to medium-high. Add three-quarters of the onion, 2 cloves garlic, 1 chopped jalapeño and the cumin.
Cook, stirring often, until the onions are soft, about 5 minutes. Add 3/4 cup water and 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 and reduce the heat to low.
Add tomatillos to a food processor along with the remaining onion, garlic, jalapeño, cilantro, mint, juice of 1 lime and 2 tablespoons water. Process until almost smooth; transfer to a bowl and season with salt.
Rinse out food processor bowl then purée half the bean mixture and juice of remaining lime to almost-smooth.
Position a rack in the center of the oven and preheat the broiler. Spread tortillas with the puréed beans, top with some of the whole bean mixture then sprinkle with cheese. Broil until cheese melts. Top with the lettuce, salsa verde and jalapeño rings, and serve.