Place a large skillet over medium heat. Add 1 tablespoon of oil followed by the onion, ½ of the jalapeno, salt, and pepper. Cook just until the onions begin to soften, about 3 minutes. Remove the pan from heat and stir in the garlic. Add the beans and mash with a potato masher or fork until smooth, yet still slightly chunky. Stir in the crushed tortilla chips, egg, and ¼ cup of the cheddar. Form the mixture into 4 equally-sized patties.
Preheat the oven to 400ºF. In a small food processor, combine the mayonnaise, the remaining jalapeno, cumin, salt, and pepper, then blend until smooth. Strain the mayonnaise mixture into a small bowl and place in the refrigerator.
Place a large skillet over medium heat. Add the remaining 3 tablespoons of oil to the pan followed by the patties and cook until browned on both sides and cooked through in the middle, about 6 minutes per side. Transfer to a plate.
Place the buns, cut side up on a baking sheet and toast until golden, about 7 minutes. Top each bottom bun with a burger, sprinkle the burgers with the remaining cheddar, and spread about 1 tablespoon of the jalapeno cream on the cut side of the top bun. Top each burger with a slice of onion, lettuce leaf, and a dressed bun, then serve.