In a large skillet over medium heat on the stove, 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 both sides and set aside.
Prepare the grill for direct cooking over medium-high heat (400° to 500°F).
Mix the ground chuck, Worcestershire sauce, 1 teaspoon of salt, and the pepper, incorporating the spices evenly. Gently shape into six oval patties of equal size, each 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 the patties to keep them from forming a dome as they cook.
Brush the cooking grates clean. Grill the patties over direct medium-high heat, with the lid closed, until cooked to medium doneness, 9 to 11 minutes, turning once. Transfer the patties to a rimmed baking sheet. Lower the temperature of the grill to medium heat (350° to 450°F).
Brush the cooking grates clean. Grill the patties over medium-high heat, with the lid closed, until cooked to medium doneness, 9 to 11 minutes, turning once. Transfer the patties to a rimmed baking sheet. Lower the temperature of the grill to medium heat (400°F).
Grill the bread slices (one side only) over direct medium heat until toasted, about 1 minute. Transfer the bread, toasted sides up, to a work surface.
Grill the bread slices (one side only) over medium heat until toasted, about 1 minute. Transfer the bread, toasted sides up, to a work surface.
Evenly divide the caramelized onions on six 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 over the sandwiches and toast the other side. Serve the patty melts warm with mustard, if desired.
Evenly divide the caramelized onions on six 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 medium heat until the bread on the bottom is toasted, about 1 minute, and then carefully turn over the sandwiches and toast the other side. Serve the patty melts warm with mustard, if desired.
