Pat chicken breasts dry and season both sides evenly with the BBQ seasoning.
In a mixing bowl, whisk together the peach preserves, soy sauce, dry mustard, minced garlic, cayenne, salt, and black pepper until smooth.
Preheat the grill or grill pan to medium-high heat.
Grill the chicken for about 6 minutes per side, flipping once, until the chicken is almost cooked through (internal temperature about 155°F/68°C).
During the final minutes of cooking, brush both sides of each breast generously with the peach sauce and continue grilling until the glaze is warmed and the internal temperature reaches 165°F (74°C).
Remove the chicken from the grill and let it rest a few minutes; serve with any remaining sauce on the side.