Combine the brisket, onion, garlic, and peppercorns in a stockpot or large saucepan. Add enough water to cover the meat by about one inch. Bring to a boil over high heat, reduce the heat to medium-low, and cook, covered, for 2 to 2 1/2 hours, or until the meat is tender when pierced with a fork. Check the meat for tenderness after 2 hours.
Lift the meat from the cooking liquid and discard the liquid. When the meat is cool enough to handle, pat dry with paper towels. Rub all sides of the brisket with the dry rub, transfer to a shallow dish, cover loosely, and set aside at room temperature for about 1 hour, or refrigerate for as long as 24 hours. Alternatively, put the brisket in a sealable plastic bag and refrigerate.
Prepare a charcoal or gas grill. Lightly spray the grill rack with vegetable oil cooking spray. The coals should be moderately hot.
Grill the brisket for about 20 minutes. Turn and cook for 20 minutes longer, or until the internal temperature is between 190° and 200°F. Let the meat rest for about 15 minutes before slicing thinly across the grain. Serve with the barbecue sauce.