Assemble sandwiches: spread mayonnaise on each bread slice; stack ham, turkey and cheese on half the slices; top all with another slice of bread.
Press each sandwich firmly.
In a shallow dish, whisk together eggs and milk.
Melt butter in a large skillet over medium heat.
Dip each sandwich in egg and milk mixture, coating completely.
Place sandwiches in skillet; brown first side, flip, brown second side.
