Pour oil into a medium skillet. Place over medium heat.
When oil is hot and shimmering, add sliced onions. Sprinkle with sugar.
Cover skillet and cook, stirring occasionally, until onions are tender and golden brown, about 15–20 minutes.
Season with a pinch of kosher salt. Remove onions from heat, Set aside.
Spread softened butter on both sides of each rye bread slice. Set aside.
In a large bowl, add ground chuck, Worcestershire sauce, 1 teaspoon of kosher salt, and ground black pepper, to taste. Mix gently to combine fully.
Divide mixture into 6 equal portions.
Shape into patties about 3/4-inch thick. Use your thumb or a spoon to press a shallow indentation, about 1 inch wide, into the center of each patty to help them cook evenly.
Preheat grill for direct cooking over high heat. Clean the cooking grates.
Grill patties with the lid closed as much as possible, turning once, until cooked to medium, about 8 - 10 minutes total.
Transfer the patties to a work surface or plate and reduce the grill’s heat to medium.
Place buttered bread slices on grill. Toast one side only over direct medium heat, about 1 minute.
Remove from the grill and place the bread, toasted sides up, on a work surface or plate.
Divide caramelized onions evenly among six slices of toasted bread.
Top each with a grilled patty.
Sprinkle generously with grated Havarti or Swiss cheese.
Place remaining slices of bread, toasted sides down, on top of the patties to form sandwiches.
Using a wide spatula, carefully return sandwiches to the grill. Toast over direct medium heat until bottom side is golden brown, about 1 minute. Gently flip sandwiches and toast the other side, another minute or so, until bread is crisp and cheese has melted.
Serve patty melts warm with Dijon or spicy brown mustard on the side.