In a large pot over high, combine 2½ quarts water, the onion, garlic, bay and peppercorns. While the water heats, use a sharp knife to make several slashes in the skin and flesh on both sides of each wing or drumstick, cutting all the way to the bone. Add the pieces to the pot as they are prepped.
When all the chicken is in the pot, cover and bring to a boil. Uncover, reduce to medium, and cook at a vigorous simmer for 35 minutes. Strain through a fine mesh strainer set over a large bowl; discard the solids or, if desired, remove the chicken and reserve for another use. If desired, season with salt.
If you used drumsticks, remove the meat, discarding the skin and bones; you’ll have about 3 cups meat for making chicken salad or adding to soups.
If you used wings, discard the tips (if present), toss the drumettes and flats with about 1 cup barbecue sauce and arrange in an even layer on a foil-lined rimmed baking sheet; broil about 4 inches from the element until well browned on both sides, 8 to 10 minutes total, flipping the wings about halfway through.
