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1
Melt 3 tablespoons butter in a large skillet (cast-iron is preferred) over medium heat.
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2
Add in onions; cook/stir until caramelized, about 20 minutes; remove onions from pan and set aside.
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3
Lay rye bread on waxed paper on a baking sheet.
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4
Butter each slice of bread on the top side, dividing 1 tablespoon of the remaining butter among 8 slices; flip bread over and butter the other sides of the 8 slices, dividing 1 tablespoon of butter among them.
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5
Reheat skillet that cooked the onions over medium heat.
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6
Toast bread on both sides in the skillet until the bread is light gold, about 1 minute per side; set aside.
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7
Divide beef into 4 even amounts; shape each into an oval patty about 1/2 inch thick.
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8
Season both sides with salt and pepper.
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9
Add to the same skillet you have been using over med-high heat; cook patties 2 1/2 to 3 minutes per side for medium-rare doneness; remove from skillet and put sandwiches together.
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10
Lay out 4 slices of toasted bread; top with 1/4 of the onion mixture, then a slice of cheese, a beef patty, and a second slice of cheese; top with remaining 4 toast slices.
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11
Melt in remaining 1 tablespoons butter in skillet; toast sandwiches on both sides, about 1 minute per side, until bread is toasty brown and cheese is melted.
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12
Serve immediately.
