Spread 4 slices with Dijon; layer each with Swiss, ham, turkey, another Swiss slice, and another bread slice.
Press gently; trim only loose meat beyond bread.
Whisk eggs, milk, and salt streak-free in a shallow dish.
Melt half the butter on a griddle over medium-low. Quickly dip 2 sandwiches on both sides; place on griddle.
Cook 3 to 4 minutes per side until golden, egg sets, and center reaches 165°F. Repeat with remaining butter and sandwiches.
Dust each with 1/2 teaspoon powdered sugar; cut diagonally and serve hot.