In a large skillet over medium-high heat, melt 2 tablespoons of butter. Add the onions and season with salt and pepper. Cover and cook, stirring occasionally, until softened, about 3 to 5 minutes. Uncover, lower the heat to medium-low, and cook, stirring occasionally, until deep golden brown, 20 to 25 minutes. If the onions look dry, add a few tablespoons of water. If desired, deglaze with vinegar and cook until evaporated, about 1 minute. Transfer to a bowl and season to taste.

Add the cheese to the onions and stir to combine. Place two slices of bread down, scoop half of the cheese-onion mixture onto each slice, and top with the remaining bread slices, pressing down gently.

In the same skillet, melt 1 tablespoon of butter over medium heat. Add the sandwiches and cook until the bottoms are golden brown, about 3 to 4 minutes. Add the remaining 1 tablespoon of butter, flip the sandwiches, press down, and cook until the cheese melts and the bottoms turn golden brown, another 3 to 4 minutes. Serve immediately.