Heat 1 teaspoon olive oil in a large nonstick skillet over medium-high heat. Add the onions and cook, stirring, until soft, 2 to 3 minutes. Push to one side of the skillet; add the mushrooms and cook until browned, about 2 minutes. Season with salt and pepper, then mix the mushrooms and onions together and cook until the onions are golden, about 10 more minutes. Add the vinegar and 2 tablespoons Worcestershire sauce. Transfer the mixture to a bowl and keep warm.
Meanwhile, mix the turkey, beef and the remaining 2 teaspoons Worcestershire sauce in a bowl. Shape into 4 equal-size oval patties.
Wipe out the skillet, add 1 tablespoon olive oil and place over medium-high heat. Season the patties with salt and pepper and cook until browned on the bottom, about 4 minutes. Flip and cook through, 1 to 2 more minutes.
Divide the onions and mushrooms among 4 bread slices; top with a slice of cheese, a patty, another slice of cheese and another bread slice. Wipe out the skillet; add 1 tablespoon olive oil and place over medium heat. Cook the sandwiches in two batches until the bread is toasted and the cheese melts, 1 to 2 minutes per side, adding the remaining 2 teaspoons oil as needed.