To make the patties, roll six 5-ounce sirloin balls and form each ball into a patty. Arrange on a tray, cover, and refrigerate.
Toast the buns. Set aside.
Heat a large skillet over medium-high heat and add just enough oil to cover the entire bottom. When the oil begins to smoke, reduce the heat to medium, and place the patties into the skillet. Season the patties with salt and pepper, and cook for 3 minutes. Flip, and cook on the other side for 1 minute more. Place 1 slice cheese on each patty and continue cooking 2 minutes more for medium-rare doneness. Cover with a lid for the last 30 seconds to melt cheese.
To assemble the burgers, place 1 patty on 1 toasted bun bottom. Spoon some salsa on each patty. Spread some of the mayonnaise on the bun top and cover. Repeat with the remaining ingredients. Dont forget to wrap the sandwiches in wax paper. Let rest for 2 to 3 minutes and serve