Melt 2 tablespoons of the butter with the vegetable oil in a large saute pan over medium-low heat. Add the onions and 1/2 teaspoon salt. Cook until the onions are golden and caramelized, 25 to 30 minutes. Set aside.
In a medium bowl, combine the beef, pork, Worcestershire, 1 teaspoon salt and 1/4 teaspoon pepper. Divide the mixture into 6 equal portions and form each into a patty in the shape of and a little larger than a slice of the rye bread.
Add some oil to a large griddle pan and set it over medium-high heat. When the pan is hot, add the burgers in batches if needed, being careful not to crowd the pan. Cook for 3 to 4 minutes on each side for medium doneness. Remove the burgers to a platter and cover loosely with foil. Repeat with the remaining burgers if needed.
Set a large skillet or griddle pan over medium-low heat (or wipe out the griddle pan from the burgers and use that). Butter one side of each slice of rye bread with 1/2 tablespoon of the remaining butter and place the bread slices butter-side down on the griddle. Lay 2 slices of cheese on 6 of the bread slices, then add a burger, one-sixth of the onions and 2 more slices of cheese on top. Place one of the remaining bread slices on top of each sandwich, buttered-side up. Cook until the bottoms are golden brown and the bottom cheese is melted, 2 to 3 minutes. Using a spatula, flip the sandwiches to toast the other sides and melt the cheese, another 2 to 3 minutes. Remove to a platter and cover loosely with foil.