Cook the Bacon: In a large skillet, heat over medium heat and add the bacon strips, cooking until they're crispy and golden brown, about 5-7 minutes. Transfer the bacon to a plate lined with paper towels to drain excess fat.

Caramelize the Onions: In the same skillet with the bacon fat, add 2 tablespoons of unsalted butter. Let it melt over low heat, add the thinly sliced onions, and stir occasionally for about 15-20 minutes, seasoning with salt and pepper until deeply golden.

Assemble the Sandwich: Lay two slices of sourdough bread on a surface. Place a layer of cheese on one slice, then the bacon and a spoonful of caramelized onions, topping with another slice of cheese and the second slice of bread.

Grill the Sandwiches: Heat the skillet again, adding more butter. Place the assembled sandwich in the skillet and grill for 3-4 minutes until golden. Flip and grill the other side until the cheese is melted and the bread is golden.

Rest and Serve: Remove from skillet, let rest for 1 minute, then slice each sandwich diagonally and serve warm.