Place a cast iron skillet, or equally heavy bottom skillet onto the stove, and then heat to medium-high.
Peel and then thinly slice 1 yellow onion into strips.
Melt 3 Tbsp butter in the warm skillet.
Add the onion slices to the melted butter and then stir to coat the onions in butter.
Season with salt and pepper if desired, and then stir with a spatula often to prevent burning.
Cook the onions until they are slightly browned and translucent.
Transfer the fried onions to a bowl and then set aside.
Make the Burger Patty's
Preheat the oven to BROIL.
Place a rimmed baking sheet on the countertop and then spray it with non-stick spray.
Place the burger patties onto the baking sheet, and then place the baking sheet into the oven and cook until the internal temperature reaches 160 degrees Fahrenheit.
Remove the baking sheet with the cooked burgers from the oven and then place it onto a potholder to cool.
Assemble the Patty Melts
Preheat a griddle to 350 degrees Fahrenheit.
Butter one side of a piece of bread and then place it onto the warm griddle butter-side-up, do this for as many pieces of bread that will fit on the griddle.
When the bread is toasted, flip the bread with a spatula and then lay a slice of cheese onto it.
Place a cooked burger patty onto the sliced cheese on the toasted bread.
Measure 1 tsp fried onions and then place them onto the patty, spreading them out evenly.
Place another piece of cheese onto the fried onions, and then using a spatula, flip a slice of toasted bread, toasted side down, onto the sliced cheese and then continue to toast the Patty Melt until both outer sides are fully toasted.
Remove the Patty Melt from the griddle with a spatula, place on a platter, and then continue to cook and plate the rest of the sandwiches.
Finally, serve with your favorite sides and enjoy!