Place a skillet over medium heat and add the butter.
Once the butter has melted, add in the sliced onions, 1/2 teaspoon of salt, and a teaspoon of sugar, which will help the carmelization process.
Let the onions cook for at least 20 minutes, or until they've reached the desired level of caramelization.
Remember to stir the onions frequently and lower the heat in case they start burning and sticking to the bottom of the skillet.
To prepare the ground beef, add a packet of Worcestershire sauce, 1/2 teaspoon of salt, and black pepper.
Make sure to mix the meat and the seasonings well, so that the seasonings are evenly distributed throughout the mixture.
Once you've created the meat mixture, it is time to form the patties. You don't want the patties to be too thick, as this could prevent them from cooking all the way through.
Add some vegetable oil into a skillet, then let it heat up over high heat. Add the patties into the skillet and let them cook for about five minutes on each side.
Avoid flipping the patties more than once, as this will prevent them from preserving all the natural juices.
Once the beef patties are cooked, it is time to toast the bread and assemble the patty melt.
Coat the bread slices with a little bit of butter on one side, which will be the outer side of the sandwich.
Add the burger patties, as well as grated cheese and caramelized onion, then top with another bread slice.
Heat up a little bit of butter in a pan, then add the patty melt sandwich, making sure that the side with the butter is down.
Toast the sandwich for about 3-5 minutes on both sides until the cheese melts, or until you're satisfied with how crispy the bread is.
Serve with mayonnaise and your favorite condiments.