Heat the olive oil in a large, heavy-bottomed saucepan on medium heat. Add the onions, stir briefly to coat with oil and cook without stirring for 10-12 minutes.
Flip so that the onions on the bottom of the pot are now on top, and cook for another 10-12 minutes. Flip one more time, lower heat and allow onions to caramelized until completely brown, about 15 minutes, gently stirring occasionally. Remove from heat and set aside. These can be reheated before you put the burgers together.
In a large bowl, mix ground beef, onion powder, salt and pepper until just combined. Do not over mix or your patties will be tough.
Divide into four portions and form patties, configure to that the burger fits the bread. Make these right before you grill them, so they stay at room temperature.
Preheat your grill, grill pan or cast-iron skillet to high heat and add burger patties. If using a grill, cover with the lid.
Cook until the crust that forms on the bottom of the burger releases it from the pan or grate, about 2 minutes. Gently test, but don't flip it until it gets to this point. When burgers lift up easily, flip and cook on the other side for another 2-3 minutes for medium to medium rare, then transfer to a plate and allow to rest.
Assemble the patty melts: spread both slices of bread for each sandwich with thousand island dressing, then add a patty, two slices of cheese and top with caramelized onions.
Preheat a cast-iron skillet and melt half a tablespoon of butter for each sandwich. Drop a sandwich in, press down with a spatula and grill until golden brown, about 2-3 minutes. Flip and grill on the other side.
Slice down the middle and serve with extra thousand island dressing on the side for dipping.