1. In a mixing bowl, whisk together the eggs and milk.
2. Assemble the sandwiches by layering a slice of ham, turkey, and cheese between two slices of bread.
3. Dip each sandwich in the egg mixture, making sure to coat both sides.
4. Heat butter in a skillet over medium heat. Cook the sandwiches until golden brown on both sides and the cheese is melted, about 3-4 minutes per side.