Place beef chunks on baking sheet in single layer, leaving 1/2 inch of space around each chunk. Freeze meat until very firm and starting to harden around edges but still pliable, 15 to 25 minutes. Place half of meat in food processor and pulse until meat is coarsely ground, 10 to 15 one-second pulses, stopping and redistributing meat around bowl as necessary to ensure beef is evenly ground. Repeat. Gently separate ground meat into 4 equal mounds. Without picking meat up, with your fingers gently shape each mound into loose patty 1/2 inch thick and 4 inches in diameter, leaving edges and surface ragged. Season top of each patty with salt and pepper. Using spatula, flip patties and season other side.
Add oil to a heavy-bottomed 12-inch skillet over high heat heat until just smoking. Using spatula, transfer burgers to skillet and cook without moving for 3 minutes. Using spatula, flip burgers over and cook for 1 minute. Top each patty with slice of cheese and continue to cook until cheese is melted, about 1 minute longer.
Whisk together in small bowl
Transfer patties to toasted bun bottoms and top with onion. Spread 2 teaspoons of burger sauce on each bun top.