Preheat the oven to 350 degrees.
In a small bowl, combine the breadcrumbs and oil. Spread on a baking sheet and toast for 3 minutes. Set aside to cool.
Cut open the side of each chicken breast to make a pocket. Place 2 slices of Swiss cheese and 2 slices of ham in each. Use toothpicks to secure it closed.
In a bowl, combine the egg and flour. Place breadcrumbs on a plate and season with salt and pepper.
Roll the chicken in the egg mixture, then breadcrumbs.
Place on sheet pan and bake 25-30 minutes, or until chicken is cooked to 165 degrees internally.
To make the creamy Dijon sauce: Melt the butter in a small saucepan. Whisk in the flour. Pour in milk and whisk, cooking until slightly thickened. Add in the mustard and parmesan cheese. Cook an additional minute. Serve on top of chicken.
