Begin by assembling the sandwiches. Layer 2 slices of bread with ham, turkey (or chicken), and Swiss cheese. Top with another slice of bread to create a sandwich. Repeat for the remaining ingredients to make 4 sandwiches total.
In a large bowl, whisk together the eggs, milk, vanilla extract, ground cinnamon, salt, and pepper until well combined.
Heat a large skillet or griddle over medium heat and add a tablespoon of butter, allowing it to melt and coat the surface.
Dip each sandwich into the egg mixture, ensuring both sides are well-coated but not overly soggy. Allow any excess egg mixture to drip off.
Place the coated sandwiches in the skillet and cook for 3-4 minutes on each side, or until golden brown and the cheese is melted. Add more butter to the skillet as needed.
Once cooked, remove the sandwiches from the skillet and let them rest for a minute. Dust with powdered sugar if desired.
Serve warm with jelly or jam on the side for dipping.