Pat the chicken completely dry. Drizzle with avocado oil and rub to coat evenly. Season all over with kosher salt, black pepper, and BBQ spice blend.
Add the peaches (with some of the juice), BBQ sauce, apple cider vinegar, garlic, shallots, Dijon mustard, smoked paprika, and olive oil to a blender. Blend until completely smooth.
Taste the sauce and adjust as needed. If it leans sweet, add kosher salt, black pepper, or a splash more vinegar until balanced.
Divide the sauce in half. Reserve one half for brushing on the grill. Transfer the other half to a saucepan.
Place the saucepan over medium heat and simmer for 5–8 minutes until slightly thickened. Keep warm.
Preheat grill to medium heat and oil the grates.
Place the chicken on the grill with no sauce. Cook undisturbed for 4–5 minutes to develop char.
Flip, then begin brushing with the reserved sauce. Continue flipping and brushing every couple of minutes, layering the glaze as it caramelizes.
Cook until the chicken reaches 165°F and is glossy with light char, about 10–12 minutes total.
Transfer to a cutting board and let rest for a few minutes.
Slice the chicken and toss lightly with some of the warm, simmered sauce. Serve with extra sauce on the side.