Preheat oven to 350°F. Place an oven-safe wire rack inside a rimmed baking sheet, and set aside.
Place chicken between 2 pieces of parchment paper; use a rolling pin to flatten to about ¼-inch thickness.
Place chicken breasts, smooth sides down, on a plate. Sprinkle evenly with ½ teaspoon of the salt and ¼ teaspoon of the pepper.
Place 3 Swiss cheese strips crosswise in the center of each chicken breast; place ham slices on top of cheese.
Fold each chicken breast in thirds, like an envelope, over ham and cheese to seal; secure with wooden picks.
Place flour, egg, and panko in 3 separate wide, shallow bowls. Stir remaining 1 teaspoon salt and ¼ teaspoon pepper into panko.
Working with 1 chicken breast at a time, dredge in flour, and shake off excess. Dip in egg, letting excess drip off. Dredge in panko to coat, pressing to adhere. Place on prepared wire rack.
Pour oil to a depth of ¼ inch in a large skillet, and heat over medium. Working in 2 batches, place breaded chicken in hot oil; fry until golden brown, 2 to 4 minutes per side. Return chicken to prepared rack.
Bake in preheated oven until chicken is cooked through, 5 to 7 minutes. Remove from oven; lightly sprinkle with additional salt. Remove and discard wooden picks. Serve chicken with Herby Dijonnaise.
