Combine eggs, milk, sugar, and salt; whisk together. Top 6 slices of bread with the ham; cover with remaining bread. Dip each sandwich into egg mixture. Heat butter in large skillet over medium-low heat. Brown sandwiches on both sides, about 5 minutes on each side. Cut sandwiches in half; arrange on hot platter.
Makes 6 sandwiches.
These can be made in a sandwich maker.