Place each chicken breast on a cutting board and carefully butterfly it by slicing horizontally through the thickest part, being careful not to cut all the way through—you want one connected piece that opens like a book. Once butterflied, place plastic wrap over the chicken and pound it gently and evenly to about 1/4-inch thickness using a meat mallet. This ensures even cooking and makes rolling easier. Lay the pounded breasts flat and ready for filling.
Working with one chicken breast at a time, lay 2 slices of ham on the flattened chicken, leaving a small border around the edges. Place 2 slices of Swiss cheese on top of the ham, then carefully roll the chicken tightly from one end, tucking in any filling as you go. Secure each roll with 2-3 toothpicks placed horizontally to prevent the roll from unraveling during grilling. Repeat with remaining three chicken breasts.
In a small bowl, whisk together the mayo, Dijon mustard, white wine vinegar, sugar, garlic powder, cayenne, salt, and cracked black pepper until smooth and well combined. I always use Hellmann's mayo because it creates the creamiest, most luxurious sauce—it really makes a difference in the final flavor. Set the sauce aside at room temperature while you prepare the grill.
Preheat your grill to 375°F. While the grill heats, rub the exterior of each rolled chicken breast evenly with the poultry rub, coating all sides. Once the grill reaches temperature, place the chicken rolls seam-side down on the grill grates and cook for 15-18 minutes, then carefully flip and cook for another 15-18 minutes on the other side. The chicken is done when it reaches an internal temperature of 165°F measured at the thickest part with a meat thermometer.
Transfer the grilled chicken to a cutting board and let it rest for 10 minutes before serving—this allows the juices to redistribute throughout the meat, keeping it moist and tender. I find this resting period makes a noticeable difference in texture, especially with stuffed breasts. Remove the toothpicks, plate the chicken, drizzle or spoon the creamy mustard sauce from Step 3 over the top, and garnish with fresh parsley.