In a large skillet over medium-high heat, melt 2 tablespoons butter. Add the onions and season with salt and pepper. Cover and cook, stirring occasionally, until deep golden brown, about 20 to 25 minutes. If dry, add a bit of water. If desired, deglaze with vinegar and cook until evaporated. Transfer to a bowl and season with salt and pepper.

Add the cheese to the onions and stir. Place two slices of bread down, scoop half the cheese-onion mixture on each. Top with remaining bread slices and press down gently.

In the same skillet, melt 1 tablespoon butter over medium heat. Add sandwiches and cook until golden brown, about 3 to 4 minutes. Flip, add remaining butter, and cook until cheese is melted and bottoms are golden, about another 3 to 4 minutes. Serve immediately.