Heat 2 teaspoons of butter in a skillet over medium heat. Add the sliced onions and stir. Let the onions caramelize, stirring very occasionally, for about 15-25 minutes. Season with 1/4 tsp salt. Set aside.
Meanwhile, season the ground beef with remaining 1/4 tsp salt, Worcestershire sauce, and black pepper. Shape into two thin patties. Heat a non-stick skillet over medium-high. Cook the patties for about 5 minutes per side until done. Set on paper towels to absorb excess grease.
When ready to make the patty melt, spread each slice of bread with 1 teaspoon of butter each. Place two slices of bread, butter side down on a plate. Sprinkle each with a little cheese. Top evenly with the onions then put the cooked beef patty on top. Sprinkle with remaining cheese, and cover with bread (butter side out).
Heat a skillet or griddle over medium-high heat. Place the sandwiches in the skillet (with the butter on the outside). Cook for about 3-5 minutes until toasted. Carefully flip and cook another 3-5 minutes until toasted and cheese has melted. Slice in half & enjoy!
Recipe from Living Well Kitchen https://blog.memeinge.com/patty-melt/