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Build a hardwood charcoal fire in the bottle of barbecue pit. Pile the coals together in the center and light. When the coals glow red and being to turn gray with ash, spread them in an even layer in the pit.
Rinse the chicken breast quarters and pat them dry using absorbent paper towels; set aside. Combine the East Texas Rib Rub with just enough olive oil to make a spreadable paste. Spread the paste liberally over the breast quarters; set aside on a baking sheet. Combine all Mustard Mop ingredients, except mustard, in a small saucepan and cook over very low heat for about 30 minutes. Remove the parsley and bay leaf. Stir in the Texas Sweet Jalapeno Mustard, blending well.
When the fire is ready, place the chicken, skin side down, on the grate. Close the lid and cook about 10 to 15 minutes, or until the skin begins to brown. Turn the chicken, close the lid and cook until the other side browns. Turn the chicken once more and baste with the mopping sauce. Grill about 10 minutes, turn, and baste again. Continue cooking, turning and basting, until the chicken has cooked for 1 hour to 1 hour and 15 minutes. When the chicken is done the juices will run clear when chicken is pierced with fork.
Remove chicken to a platter and cover to keep warm. Return the remaining mop sauce to low heat and bring to a boil. Add any juices that have accumulated on the platter with the chicken. Taste and season with salt and pepper if needed. Pour the sauce over the chicken and serve hot.
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