In a bowl, combine the shredded cheddar cheese, mayonnaise, diced pimentos, garlic powder, onion powder, salt, and pepper. Add a dash of hot sauce if desired. Mix until well combined. Set aside or refrigerate if making ahead.
Prepare the Beef Patties:
Divide the ground beef into 4 equal portions and form into patties. Season both sides of each patty with salt and pepper.
Heat a skillet or griddle over medium-high heat. Add a little butter or oil, then cook the patties for about 3-4 minutes on each side, or until they reach your desired level of doneness. Remove the patties from the skillet and set aside.
Caramelize the Onions:
In the same skillet, add a bit more butter if needed and reduce the heat to medium. Add the thinly sliced onions and cook, stirring occasionally, until they are soft, golden, and caramelized. This may take about 15-20 minutes. Remove from the skillet and set aside.
Assemble the Patty Melts:
Butter one side of each bread slice. On the non-buttered side, spread a generous amount of pimento cheese. Place a beef patty on four of the slices, then top with caramelized onions (and any other additional toppings you like). Cover with the remaining bread slices, buttered side up.
Cook the Patty Melts:
Heat the skillet or griddle over medium heat. Place the assembled patty melts in the skillet, pressing down slightly with a spatula. Cook until the bread is golden brown and crispy, about 3-4 minutes per side, and the pimento cheese is melty.
Serve:
Once cooked, remove the patty melts from the skillet and let them rest for a minute before cutting. Serve warm, cut in half if desired.