Preheat oven: Preheat oven to 375-400°F (190-200°C).
Make egg mixture: In a bowl, whisk together eggs, milk and a pinch of salt.
Assemble sandwiches: Arrange 6 slices of bread on a flat surface. Place a slice of cheese on each slice of bread, top each with a slice of ham, then another slice of cheese, and top with remaining slices of bread to form sandwiches.
Dip and arrange: Briefly dip each sandwich in egg mixture, making sure both sides are coated. Arrange dipped sandwiches in a baking dish.
Add toppings: If desired, sprinkle additional cheese on the buns for extra cheesy flavor. Bake: Bake in the preheated oven for 15-20 minutes, or until cheese is melted and bubbly and bread is golden brown. Service: Let the dessert cool for a few minutes before cutting and serving.