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. Transfer meat to baking sheet, overturning bowl and without directly touching meat. Repeat grinding with remaining meat. Spread meat over sheet and inspect carefully, discarding any long strands of gristle or large chunks of hard meat or fat.
Gently separate ground meat into 4 equal mounds. Without picking meat up, with your fingers gently shape each mound into loose patty 1/4 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. Refrigerate while toasting buns.
Melt 1/2 tablespoon butter in heavy-bottomed 12-inch skillet over medium heat until foaming. Add bun tops, cut-wide down, and toast until light golden brown, about 2 minutes. Repeat with remaining butter and bun bottoms. Set buns aside and wipe out skillet with paper towels.
Return skillet to high heat; add oil and heat until just smoking. 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.
Transfer patties to bun bottoms and top with onion. Spread 2 teaspoons of burger sauce on each bun top. Cover burgers and serve immediately.
Classic Burger Sauce: Mix ingredients together (makes 1/4 cup).