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1
Melt the butter in a large nonstick frying pan over medium heat until foaming.
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2
Add the onions and cook, stirring rarely, until deep golden brown and caramelized, about 40 minutes.
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3
Season generously with salt and pepper, stir to combine, and transfer to a medium bowl; set aside.
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4
Wipe out the pan with paper towels and set aside.
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5
While the onions are cooking, place the meat in a large bowl and add the measured salt and pepper.
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6
Mix with your hands until just evenly combined.
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7
Divide into 4 equal portions and place on a baking sheet.
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8
Using your fingertips, press each portion into a 1/4-inch-thick patty that is slightly wider in diameter than the slices of bread.
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9
Heat a separate large frying pan (not the nonstick pan) over medium-high heat until a drop of water sizzles and immediately disappears on the surface.
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10
Place 2 of the patties in the pan and cook undisturbed until browned on the bottom, about 2 to 3 minutes.
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11
Flip and cook until the patties are just slightly pink in the center for medium, about 2 to 3 minutes more.
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12
Transfer the patties to a large plate and tent loosely with aluminum foil.
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13
Repeat with the remaining 2 patties; set aside.
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14
On 1 side of each bread slice, spread about 1 heaping teaspoon of the mayonnaise in an even layer and all the way to the edges.
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15
Place 4 of the bread slices on a work surface mayonnaise-side down, and set the other 4 slices aside.
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16
Distribute half of the cheese among the 4 bread slices and divide the cooked onions evenly over the cheese, arranging the onions in an even layer.
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17
Top with the beef patties, divide the remaining cheese over the patties, and place the remaining 4 slices of bread on top mayonnaise-side up.
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18
Heat the reserved nonstick pan over medium heat until a drop of water sizzles and immediately disappears on the surface, about 3 minutes.
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19
Place 2 of the sandwiches in the pan and cook until the bottom slices of cheese start to melt and the bread is golden brown, about 3 to 4 minutes.
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20
Flip and cook until the other side is golden brown and all of the cheese has melted, about 3 to 4 minutes more.
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21
Remove to a serving plate and repeat with the remaining 2 patty melts.
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22
Serve immediately.
