In a large skillet over medium heat, warm the oil. Add the onions, sprinkle with the sugar, cover, and cook until the onions are tender and golden brown, 15 to 20 minutes, stirring occasionally. Season to taste with salt. Remove from the heat.
Butter the bread on each side and set aside.
Prepare the grill for direct cooking over high heat (450° to 550°F).
In a large bowl gently mix the ground chuck with the Worcestershire sauce, 1 teaspoon of salt, and the pepper, incorporating the spices evenly. Gently shape into 6 patties of equal size and thickness, about ¾ inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of each patty.
Brush the cooking grates clean. Grill the patties over direct high heat, with the lid closed as much as possible, until cooked to medium, 8 to 10 minutes, turning once. Transfer the patties to a work surface.
Lower the temperature of the grill to medium heat. Grill the bread slices over direct medium heat until toasted on one side only, about 1 minute. Transfer the bread, toasted sides up, to a work surface.
Evenly divide the caramelized onions on 6 of the toasted bread slices and top each with a patty. Scatter the cheese over the patties. Place the remaining bread slices, toasted sides down, on top of the patties. Using a wide spatula, carefully place the patty melts back onto the cooking grate and grill over direct medium heat until the bread on the bottom is toasted, about 1 minute, and then carefully turn the sandwiches and toast the other side. Serve the patty melts warm with mustard, if desired.