Season ground beef with salt and pepper and divide into 3 equal-sized balls. Shape each ball as close as possible to the shape of the bread. Make an indentation in the center of each patty with your thumb.
Melt butter in a skillet over medium heat and cook onion until soft and translucent, 5 to 8 minutes. Remove from the skillet and divide into 3 equal portions. Place beef patties into the same skillet and cook until desired doneness, 3 to 5 minutes per side.
Meanwhile, spread 1 teaspoons mayonnaise on one side of each slice of bread.
Place bread mayonnaise side-down in the skillet. Top with 1 slice of cheese, 1 beef patty, 1/3 of the onions, a second slice of cheese, and another slice of bread, mayonnaise side up. Cook bread until golden brown and cheese has started to melt, 3 to 4 minutes. Flip each sandwich over and cook an additional 3 to 4 minutes.
Putting an indentation on the beef patties helps them from puffing up.
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