Divide and spread 2 tablespoons butter evenly among 1 side of each bread slice; set aside 3 slices.

Divide and spread Dijon mustard evenly on top butter on 3 bread slices; top with 4 ham slices. Set aside.

Divide and spread mayonnaise evenly on top butter on remaining 3 bread slices; top with 2 slices Swiss cheese. Press ham and cheese sides together to make sandwiches.

Whisk eggs, flour, water, baking powder, and salt together in a shallow rimmed dish; set aside.

Heat remaining 2 tablespoons butter and oil in a large skillet over medium heat. Dip both sides of each sandwich in egg mixture; add to skillet. Cook until browned on both sides, flipping once or twice.