Heat grill to high. Form meat into 4 (8-ounce) burgers and season each burger on both sides with 1/4 teaspoon of salt and a 1/4 teaspoon of pepper. Grill until charred on both sides and cooked to desired doneness.
Place 2 slices of the cheese on each burger, close the lid or tent with foil, and cook until the cheese has just melted, about 1 minute. Place burgers on buns and top with a few tablespoons of Green Chile Sauce and lettuce, tomato, and pickled jalapeno, if desired.
Place all ingredients in a blender; blend until smooth and serve with the cheeseburgers.