Lightly whisk egg, milk and a generous pinch of salt in a flat bowl.
Spread 2 slices of bread with mayonnaise.
Top with ham, then cheese, then sandwich with remaining slices of bread. Press down firmly.
Melt butter in a skillet over medium heat.
Dip each side of the sandwiches briefly in the egg mixture, then place in skillet.
Cook the first side, pressing down with an egg flip, for 3 minutes or until deep golden brown.
Flip and cook the other side for 3 minutes, pressing down firmly with the egg flip, until golden brown and the cheese is melted.
Serve immediately, while hot!
