Assemble two sandwiches by layering cheese, ham, and cheese between bread slices.
Whisk eggs, milk, salt, and pepper in a shallow bowl.
Dip each sandwich in the egg mixture, coating both sides evenly.
Melt butter in a skillet over medium heat.
Cook one sandwich 3–4 minutes per side until golden and cheese melts.
Repeat with the second sandwich.
Rest briefly, then slice diagonally and serve hot.