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Cook the Onions
1. Warm olive oil in a skillet over medium heat.
2. Slice onions into thin rings and add to the skillet. Cook until softened, 5 to 7 minutes.
3. Add salt, stir, and then add wine. Cook until liquid evaporates.
4. Reduce heat, add butter, and cook until onions are golden brown, 10 to 15 minutes. Remove from heat and transfer to a plate.
Make the Patty Melt
1. Wipe out the skillet and add enough oil to coat the surface over medium heat.
2. Shape ground beef into patties, season with salt, and cook until browned, 3 to 5 minutes per side for medium-rare.
3. Place burgers on a wire rack. Butter one side of each bread slice.
4. In the skillet, place a slice of bread buttered side down, add cheese, patty, onions, more cheese, and another bread slice (buttered side up).
5. Cover and cook for 2 1/2 minutes, then flip and cook for 1 more minute.
6. Remove the patty melt from the skillet and repeat with remaining ingredients. Devour hot
