Assemble sandwiches by buttering the inside of the bread and arranging ham and cheese evenly to the edges with a slice of bread between the ham layer and the cheese layer.
Combine dijon and mayonnaise and spread onto the interior ham side of one slice of bread.
Whisk egg, add milk, salt and pepper.
Heat a wide skillet, add butter and melt. Dip both sides of sandwich in egg mixture and grill cheese side down until golden brown and cheese is melting. Press down with spatula to help bread make full contact with hot pan and to ensure the inside of the sandwich heats through.
Flip and grill on other side, pressing again with spatula.
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