For brine: Combine water, salt, brown sugar, and thyme in a plastic storage bag, stainless steel bowl, or any vessel large enough to ensure chicken is covered with the brine. Stir to dissolve salt and sugar.
Add chicken to the liquid. Make sure all of the chicken pieces are covered with water. Allow chicken to brine in the refrigerator for 2 hours. If you are short on time, still do this step, even if it is only for 20 minutes. (This process will add flavor, tenderize the meat and help it stay moist during cooking. The salt in the solution penetrates the meat and breaks down the proteins, causing the chicken to be more tender)
For mustard sauce: Combine mustard, apple cider vinegar, water, brown sugar, honey, lemon juice, Worcestershire, cayenne pepper, and butter in a sauce pan. Simmer for 30 minutes, whisking every 5 minutes or so.
Remove chicken from brine and pat dry with paper towels. Grill over medium heat 20 minutes, turning half way through cooking time.
Move chicken off direct heat to the other side of the grill and brush with mustard sauce. Continue cooking 20 more minutes or until internal temperature reaches 155 degrees and juices run clear
Serve with additional mustard sauce.