Preheat oven to 400°F. Set a wire rack on a lined baking sheet.
Set up three shallow dishes with flour, beaten eggs, and seasoned breadcrumb mixture.
Bread each chicken breast through flour, egg, then breadcrumbs, pressing firmly to coat. Place on the wire rack.
Drizzle lightly with olive oil or melted butter. Bake for 20 minutes until golden and internal temperature reads 155°F.
Top each breast with one slice of ham and one slice of Swiss cheese. Return to oven for 4 to 5 minutes until cheese melts.
Whisk Dijon mustard, honey, mayonnaise, and apple cider vinegar together for the sauce. Serve alongside the chicken.