Combine the ground beef with 2 tablespoons of butter, Worcestershire sauce, salt, and pepper. Mix until well blended and shape into 4 large patties. Make an indentation in the center of each burger patty. This will help them cook evenly and will prevent them from puffing up in the middle as they cook.

Melt 2 tablespoons of butter in a large skillet over medium heat. Add the diced onion and cook, stirring, for 7 minutes or until they are tender and translucent. Remove the onions to a plate and set aside.

Preheat the broiler if you plan to toast the buns in the oven. 

Wipe out the skillet and place it over medium-high heat. Cook the burger patties for about 3 minutes per side or until cooked through. Place a slice of cheese on each burger and cook for a few seconds longer to melt the cheese.

Meanwhile, place the buns on a baking sheet, cut side up, and toast under the broiler. Watch carefully to avoid burning, as broiler heat does vary. Alternatively, toast the buns on a griddle on the stovetop.

Place a few lettuce leaves on a toasted bun and top with a burger patty and a generous amount of onions. Slather the top of the bun with 1 1/2 tablespoons of butter and place it on the hot burger and onions.

Serve immediately. Enjoy!