Preheat a 10 or 12-inch heavy-bottomed skillet over medium-high heat. Add 1 tablespoon butter and swirl until foaming. Add onions, 1/2 teaspoon salt and a 1/4 teaspoon pepper and 2 tablespoons water to skillet. Cook onions, stirring often, until water evaporates and onions begin to brown, about 3 minutes. Add 2 more tablespoons water and scrape up any browned bits from bottom of skillet. Cook, adding a little water and stirring, until onions are soft and deep golden brown, about 20 minutes total. Remove to a plate and wipe out skillet with a paper towel.

Return skillet to high heat. Season burger patties on both sides with salt and pepper. Add 1 tablespoon butter and swirl until melted. Add burger patties and cook, without moving them, until a dark brown crust forms on the first side, about 2 minutes. You can reduce the heat a little if butter begins to smoke excessively. Flip burgers and cook on the second side until crust develops, 1 1/2 minutes longer. Transfer the patties to a plate and wipe out the skillet again.

Spread 2 generous tablespoons of the kimchi pimento cheese onto one side of each of the 4 slices of bread. Divide the caramelized onions on top of the cheese. Add burger patties to two of the slices and close sandwiches.

Add 1 tablespoon of butter to the skillet. Return heat to medium and swirl the pan until the butter melts. Add the sandwiches and cook, moving them around the pan, until deep golden brown on the bottom, about 5 minutes. Remove sandwiches from skillet, melt remaining tablespoon butter in pan. Return sandwiches to skillet and cook until second side is golden brown and cheese is melted. Serve immediately.