In a large skillet over medium-high heat, melt 2 tablespoons of the butter. Cook, stirring occasionally, until the butter foams, 1 to 2 minutes.

Add the onions and season with salt. Let the onions sit, uninterrupted, for 1 minute. Continue to cook, stirring frequently, until the onions have released some liquid and are translucent, about 5 minutes.

Reduce the heat to medium-low and cook for 30 to 45 minutes, stirring occasionally, until the onions are dark golden brown.

Transfer the onions to a bowl and carefully wipe out the skillet.

Divide the ground beef into 4 patties that are slightly wider than the slices of bread. Generously season both sides with salt and black pepper.

Return the skillet to the stove, add 2 tablespoons of the butter, and melt over medium-high heat.

Place the patties in the skillet and cook for 2 ½ minutes. Flip and cook for another 2 ½ minutes, then transfer the patties to a plate and allow them to rest for a few minutes.

Carefully drain and discard the fat, and wipe out the skillet. Return the skillet to the stove and heat over medium.

Add the remaining 2 tablespoons of butter. Once melted, add 4 slices of bread. Top the bread with one slice of cheese, a patty, some caramelized onions, another slice of cheese, and another slice of bread.

Use a spatula to press down on the patty melts. Cook until the bread is crisp, 1 to 2 minutes. Flip and press down again. Cook until toasted and crisp, 1 to 2 minutes more. Serve hot.