Step 1: In a shallow dish, whisk together the eggs, milk, salt, and black pepper until well combined. Set aside.
Step 2: Assemble the sandwiches: Lay two slices of bread on a clean surface. On one slice, layer two slices of ham, two slices of turkey, and one slice of Swiss cheese. Top with the remaining two slices of ham and two slices of turkey. Place the second slice of bread on top to complete the sandwich.
Step 3: Carefully dip each sandwich into the egg mixture, ensuring both sides are evenly coated but not overly saturated. Let any excess drip off.
Step 4: Heat a large non-stick skillet or griddle over medium heat. Add 1 tablespoon of butter and let it melt and coat the pan.
Step 5: Place one coated sandwich in the skillet. Cook for 3-4 minutes per side, or until golden brown and the cheese is melted and gooey. Remove from heat and set aside.
Step 6: Add the remaining 1 tablespoon of butter to the skillet and repeat the process with the second sandwich.
Step 7: Serve immediately. For a classic experience, dust with powdered sugar and serve with a side of raspberry or lingonberry jam.