Position a rack in the middle of the oven; preheat to 400 degrees.
In a small skillet, heat the oil, one turn of the pan, over medium-high. Add the onion, garlic, jalapeno and cumin; season with salt. Cook, stirring often, until the vegetables soften, about 5 minutes. Add 1/2 cup water. Cook, stirring often, until the liquid is reduced by half, 3 to 5 minutes; transfer to a food processor. Add the beans and hot sauce. Puree; season with salt.
Arrange the chips on a parchment- or foil-lined baking sheet; bake until toasted, 5 to 7 minutes. Top with the bean dip and sprinkle evenly with the cheese.
Preheat the broiler. In a medium bowl, combine the tomatoes, tomatillos, onion, cilantro and mint, jalapenos and lime juice; season with salt.
Return the nachos to the oven and broil until the cheese melts and is browned in spots, 4 to 5 minutes. Top the nachos with the pickled jalapenos, then the lettuce and the tomato-tomatillo salad. Serve the nacho salad on the baking sheet or divide among plates.