Combine the tomato sauce, brown sugar, mustard, and vinegar in a small saucepan. Bring the sauce to a boil and then decrease the heat to simmer. Cook, stirring occasionally, for 30-45 minutes until thickened.
Preheat the grill to medium. Next, trim excess fat from the chicken pieces amd remove the skin if desired. Wipe the chicken with damp paper towels; discard the towls immediately to prevent cross contamination. Season with salt and pepper. Place the chicken on the grill and cook until the chicken is tender and no longer pink: 10-15 minutes per side for breasts and thighs, 8-12 minutes per side for drumsticks and wings.
Finally, pour half of the sauce into a bowl to pass at the table; set aside. Baste the chicken with the remaining sauce only during the last few minutes of cooking, to prevent the sauce from burning.