Spread mustard evenly on 1 side of 6 bread slices. Top with 1 slice each of cheeses and ham; top with remaining 6 bread slices.
Whisk together eggs and next 3 ingredients until blended. Dip both sides of each sandwich in egg mixture.
Melt 2 tablespoons butter in a heavy nonstick skillet over medium-low heat. Add 3 sandwiches, and cook 3 to 4 minutes on each side or until bread lightly browns and cheese begins to melt. Repeat procedure with remaining 2 tablespoons butter and 3 sandwiches; serve immediately.