Prepare grill. Soak mesquite chips in water to cover.
MOP SAUCE: Cook bacon in medium skillet over medium heat until translucent, about 2 to 3 minutes. Add onions and continue cooking until onions are lightly browned, about 5 minutes. Transfer bacon and onions to medium saucepan; add beef broth, oil, lemon, thyme, salt and pepper. Simmer over low heat to blend flavors, 15 minutes. Set aside.
DRY RUB: While sauce is simmering, combine all ingredients in bowl. Rub dry run over entire brisket until coated. Sprinkle brisket on both sides with sugar, then vinegar.
TO COOK: When coals are covered with white ash, add wood chops. When ash begins to form on the wood chips, close vents on grill half way. Place small pan filled with water on one side of grill bottom. Place the brisket on grill directly over drip pan. Cook covered 2 hours, turning once halfway through. Continue cooking covered 2 hours more, basting with mop sauce and turning every 30 minutes. (To keep the grill at the proper temperature, replenish with additional hot coals and wood chips as needed, usually when the meat is turned.).
Preheat oven to 250°F Wrap brisket tightly in heavy foil and place in a roasting pan in the oven. Bake 6 hours or until tender enough to cut with a fork. Carve meat in thin slices across grain to serve.