Place buns on a cookie sheet cut side up. Toast them under the oven broiler; remove and set aside. (Watch closely because they brown fast)
Divide burger into 4 equal portions; form into 4 patties about ½ inches thick. Generously season patties with salt and pepper, place on a platter and refrigerate for up to 30 minutes.
Place 1 tablespoon of butter, water, and onions in a small skillet over medium heat, season with a ¼ teaspoon of salt and cook covered until onions are translucent. Remove lid and continue cooking until onion is tender and starts to brown; cover and reduce heat to warm.
Spread 2 tablespoons of butter over the top cut side of each bun.
Place cooking oil into a large skillet over high heat. When skillet starts to smoke, place burgers in a skillet, cook burgers without turning for 3 minutes, turn burgers and continue cooking for 1 more minute, top with cheese and continue cooking until cheese is melted.
Divide onion into 4 equal portions – place burger on the bun, onion, and top with the buttered top half of the bun.