Creating a classic patty melt is straightforward if you follow these steps:

1. Cook the Onions: In a skillet over medium heat, melt 1 tablespoon of butter. Add the sliced onions and cook, stirring frequently, until they are soft and caramelized (about 10-15 minutes). Season with a pinch of salt.

2. Shape the Patties: While the onions are cooking, divide the ground beef into two equal parts and shape them into burger patties. Season both sides with salt and pepper.

3. Cook the Patties: In another skillet over medium-high heat, add the remaining tablespoon of butter. Cook the patties for about 3-4 minutes on each side for medium doneness. Before removing, place a slice of cheese on each patty and cover until melted.

4. Construct the Sandwich: Spread butter on one side of each slice of bread. On the unbuttered side of the bread, layer one slice with a cheesy patty, followed by a generous helping of caramelized onions. Top with another slice of bread (buttered side up).

5. Grill the Sandwich: Place the sandwich in the skillet used for the patties over medium heat. Grill until golden brown (about 3-4 minutes), then flip carefully and grill the other side until toasted and the cheese is fully melted.

6. Slice and Serve: Remove the sandwich from the skillet, letting it rest for a minute before slicing in half. This will help keep the cheese nicely melted and contained.