Note: If using breasts, cover with plastic wrap and lightly pound the chicken with a rolling pin. This will break the membrane and help tenderize.
For the marinade: In a bowl add the marinade ingredients and mix well. Pour the contents in a ziploc bag and add the chicken. Make sure the chicken is covered with the marinade and let sit in the fridge for at least 1 hour.
For Dry Seasoning: Lightly coat all sides with olive oil. Generously season with the Mesquite, or substitute.
Steam method: Place the chicken breast or thighs on a sheet of aluminum foil, and top each piece with a pat of butter. Seal up the foil.
Place the foil packet on a 350 degree F. grill over direct heat. Close the lid and cook until the chicken reaches145 degrees F. Remove the chicken from the foil, reserve the juices, and place over direct heat. Grill for about 4 to 5 minutes, flipping occasionally, or until the internal temperature is 165 degrees F and basting with the leftover juices occasionally.
For the Direct Grilling Method: Place the chicken on the indirect heat side of the grill. Shut the lid and cook for about 5 minutes, each side or until it reaches 145 degrees F. flipping occasionally. Move the chicken over the direct heat and continue to cook for about 3 to 4 minutes, or until the internal temperature is 165 degrees F.
Optional: you can baste the chicken when it is finishing over the direct heat with BBQ sauce, if desired.