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!
This is great made with artisan (e.g. sourdough) bread or ordinary sandwich bread. Sourdough yields a slightly crunchier golden crust whereas sandwich bread has a higher ratio of soft gooey melted cheese and hot ham. I enjoy both!