Season the beef with salt and pepper. Shape it into 4 even balls.
Heat the griddle to medium-high. Place each beef ball onto the griddle, leaving plenty of space between each one. Use a smasher or strong spatula to press each one into a ½-inch-thick patty.
Cook, flipping once, until the patties are seared on each side and just cooked through, about 5 minutes per side.
Spread one side of each slice of bread with 1 teaspoon of mayonnaise.
Reduce the griddle heat to medium-low. Place four slices of the bread mayo-side down on the griddle. Top with the beef patties and a slice of cheese. Place the second slice of bread mayo-side up on top.
Toast the sandwiches until golden brown and the cheese is melted, 1 to 2 minutes per side.
Serve immediately.
Notes
Let it cook--Resist the urge to flip your patty frequently. Let it cook until it develops a nice browned crust, then flip just once.
Add onions--To make it a true classic, cook onions alongside your beef patties and add them to the sandwich, on top of the cheese.
Make it extra cheesy--Add an extra slice to each sandwich.