Preheat your Blackstone griddle over medium-high heat, around 400°F.
Place the beef patties on the griddle and cook for 3-4 minutes on each side, or until they reach your preferred doneness. You can smash the patties with a spatula for a crispy texture.
While the patties are cooking, add a small amount of butter to the griddle and sauté the onions until golden and caramelized, about 5-7 minutes.
Place the buttered bread, buttered side down, on the griddle to toast until golden brown, about 2 minutes per side.
Once the patties are cooked and the bread is toasted, place one slice of cheese on each patty. Place the patty on one slice of the toasted bread, and top with a generous portion of caramelized onions. Add the second slice of bread on top, toasted side up.
Grill the sandwich on the griddle for 2-3 minutes per side, pressing down slightly with a spatula, until the cheese is melted and the bread is golden and crispy.
Remove the patty melt from the griddle, slice it in half, and serve hot with optional ketchup and mustard for dipping.