Make the French toast: Whisk together the eggs, milk and vanilla in a medium bowl until smooth. Place the bread slices in the bowl of egg mixture and coat both sides. Heat 2 skillets (each large enough to hold 2 slices of bread) over medium heat. Add half of the butter to each. When the butter begins to foam and brown slightly on the edges, arrange 2 slices of bread in each skillet. Cook until the undersides have browned, 3 to 5 minutes. Turn the slices over and cook for an additional 3 to 5 minutes, reducing the heat to medium-low if the bread is browning too quickly. Transfer the French toast to a plate and let cool slightly. Wipe out the skillets. Preheat the oven to 350 degrees F.
Cook the sandwiches: Return the skillets to medium heat and add about 1 tablespoon butter to each. Allow the butter to melt and froth, swirling to coat the bottoms of the skillets. Divide the eggs between the skillets and spread evenly on the bottoms. Arrange two slices of the French toast, slightly apart, in each skillet. Cook, shaking the skillets a little bit so the eggs don't stick, until the eggs on the bottom of the skillet sear and firm up, 1 to 2 minutes.
Finish: Divide and spread the remaining butter on the tops of the 4 slices of French toast. Slide a spatula underneath the eggs and flip everything over so the buttered French toast slices are now on the bottoms of the skillets. Season with salt. Layer on top of one piece of French toast per skillet with the cheese, ham and turkey. Slide the spatula under one slice of the French toast and fold it over the other so it looks like a sandwich! Repeat with the second skillet. Transfer the skillets to the oven until the cheese melts and the centers get hot, about 3 minutes.
Transfer the sandwiches to a cutting board and cut each in half. Serve two halves on each plate with maple syrup for drizzling and dunking.