Preheat the oven to 400°F and line a baking sheet with parchment paper. Place each chicken breast on a cutting board, cover with plastic wrap, and use a rolling pin or meat mallet to pound them to an even 1/2 inch thickness.
Lay 2 slices of ham and 1 slice of swiss cheese on top of each chicken breast. Roll the chicken up tightly from one end to the other, tucking in the sides as you go. Secure each roll with toothpicks if needed.
Set up a breading station with three shallow dishes. Place the gluten free flour in the first dish. Whisk together the eggs and dijon mustard in the second. In the third, combine the gluten free panko, parmesan cheese, garlic powder, paprika, salt, and pepper.
Roll each chicken bundle in the flour, shaking off the excess. Dip it into the egg mixture, letting any excess drip off, then press it into the panko mixture, coating all sides evenly.
Place the breaded chicken rolls seam-side down on the prepared baking sheet. Mix together the olive oil and melted butter, then drizzle it evenly over the tops of the rolls.
Bake for 28 to 32 minutes, until the coating is golden and crispy and the internal temperature reaches 165°F. Let the chicken rest for 5 minutes, then remove the toothpicks, slice each roll in half on the diagonal, and serve.