Heat butter in a large skillet over medium-low heat. Add the onions and cook, stirring occasionally, for 30-40 minutes, until very tender. Season to taste with salt and pepper.
Burger Patties
When the onions are about halfway done, get started on the patties.
Combine ground turkey, Worcestershire sauce, salt and pepper in a large bowl and mix until just combined. Divide the mixture into 4 even parts and make large, thin patties (you want to try and get them about the size of your bread).
Heat a large skillet over medium-high heat and melt the butter. Add the patties and cook 3-4 minutes, then turn over and cook another 3-4 minutes, until patties are just cooked through. (They’ll continue to cook off the heat and are also going back in the pan as part of the sandwiches.)
Sandwiches
Place a piece of Swiss cheese on a slice of bread. Top with one of the patties then a scoop of caramelized onions. Add another slice of Swiss cheese and then the top slice of bread. Repeat for the remaining sandwiches.
Heat a large skillet over medium heat and melt the butter. Add the sandwiches (work in batches if needed) and cook for 2-3 minutes, until well browned and the cheese is melted. Flip the sandwiches over and cook on the other side until well browned and the cheese is melted. Serve immediately.