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Butterfly the chicken breasts. Place the chicken breast on a cutting board and press down on the breast with the palm of your hand. Slice into the long side of the chicken breast, parallel to the cutting board, but not all the way through to the other side. Open the chicken breast like a “book”. Place a piece of plastic wrap over the chicken breast and gently pound it with a meat mallet to make it evenly thick.
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Season the chicken with salt and pepper. Spread the Dijon mustard on the inside of each chicken breast. Layer one slice of cheese on top of the mustard, then top with the 2 slices of ham and the other slice of cheese.
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Starting with the long edge of the chicken breast, roll the chicken up to the other side. Secure it shut with 1 or 2 toothpicks.
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Pre-heat the air fryer to 350ºF.
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Set up a dredging station with three shallow dishes. Place the flour in the first dish. Place the beaten egg in the second shallow dish. Combine the panko breadcrumbs and Parmesan cheese together in the third shallow dish. Dip the stuffed and rolled chicken breasts in the flour, then the beaten egg and then roll in the breadcrumbs-cheese mixture to cover on all sides. Press the crumbs onto the chicken breasts with your hands to make sure they are well adhered. Spray the chicken breasts with olive oil and transfer to the air fryer basket.
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Air-fry at 350°F for 14 minutes, flipping the breasts over halfway through the cooking time. Let the chicken rest for a few minutes before removing the toothpicks, slicing and serving.