Divide the beef into 8 portions and flatten each between parchment into a 5-inch round.
Place one cheddar piece and 1 tablespoon jalapeños in the center of 4 rounds, leaving a 1-inch border.
Cover with the remaining beef rounds, pinch every edge firmly and smooth the seams closed.
Season the stuffed patties with salt and pepper and chill them for 10 minutes.
Cook in a heavy skillet over medium heat for 5 to 6 minutes per side, turning once without pressing.
Confirm the beef beside the cheese pocket reaches 160°F, then rest the patties for 3 minutes.
Toast the buttered buns, spread with mayonnaise and assemble with lettuce, tomato and one stuffed patty.
