Melt 2 tablespoons of the butter in a large frying pan over medium heat. Add the onion slices and a pinch of salt and cook, stirring occasionally, until the onions are evenly golden brown and tender, about 20 minutes. In a medium bowl, combine the turkey, chopped onion, Worcestershire sauce, and mustard. Season with salt and pepper. Use your hands to gently combine. Divide the meat into 4 equal portions. Shape each portion into an even 1/2-inch-thick patty. Using your thumb, make a shallow indentation in the center of each patty (this helps them cook evenly). Transfer the patties to a plate and set aside. Set a grill pan over medium-high heat. When it’s hot, spray with cooking spray. Use a metal spatula to transfer the turkey patties to the pan and cook, undisturbed (don’t be tempted to press down with the spatula!), until grill marks appear on the bottom, about 5 to 6 minutes. Flip the patties and continue cooking until they’re cooked through and an instant-read thermometer registers 165°F, about 6 minutes more. Spread the remaining 2 tablespoons of butter on one side of each bread slice. Arrange the slices buttered-side down on a work surface. To each, add a slice of cheese, a turkey patty, a quarter of the caramelized onions, a second slice of cheese, and a second slice of bread, buttered-side up. Place a large two-burner griddle or large frying pan over medium heat until a drop of water sizzles and immediately disappears on the surface, about 3 minutes. Place 2 of the sandwiches on the grill or in the pan and cook until the bottom slices of cheese start to melt and the bread is golden brown, about 3 to 4 minutes. Flip and cook until the other side is golden brown and all of the cheese has melted, about 3 to 4 minutes more. Remove to a serving plate and repeat with the remaining 2 patty melts. Serve immediately.