Place beef chunks on baking sheet in single layer, leaving ½-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 discard 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 1/2 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. Refrigerate for at least 10 minutes.
Place large cast iron skillet over high heat; add butter and heat until foaming. 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. Remove burgers from the skillet and serve immediately on soft hamburger buns and toppings and condiments of choice.
Notes
Note #1: If you go to a butcher or even have a great meat department at your local grocery store, you can ask them to coarsely grind the meat for you and skip steps #1 and #2. A great time saver!Note #2: You can shape the patties ahead of time and freeze them, separated by squares of parchment paper and wrapped in plastic wrap, then placed in a freezer ziploc bag, for up to 2 weeks. Separate the burgers and thaw at room temperature for 30 minutes before cooking.Note #3: If you don't have a cast iron skillet, use a heavy-bottomed 12-inch skillet.Note #4: If you plan to make bacon cheeseburgers, fry the bacon in the skillet first, then cook the burgers in the bacon grease instead of the butter. My mother-in-law has done this before and it's DELICIOUS.Nutritional values are based on one burger patty with cheese