Brine

Combine all ingredients in a large saucepan over medium-high heat. Cook, stirring constantly, until salt and sugar are completely dissolved. Remove from heat and cool completely (never add uncooked chicken to warm brine.)

Add chicken pieces and submerge completely; top with a small saucer with a weight on top if necessary to keep chicken submerged. Refrigerate for at least 4 hours and up to 24 hours.

Sauce

Place onion, garlic, and carrot in the bowl of a food processor; pulse, scraping down the sides, until finely chopped.

Melt butter in a skillet over medium heat; add chopped mixture and cook, stirring, until very soft, 10 to 12 minutes. Do not brown. Stir in chile powder and cook, stirring, for 2 minutes. Remove from heat and add smoked paprika; cook, stirring, for about a minute to give the paprika time to absorb some of the oil.

Add remaining ingredients except for salt, pepper, and cayenne, return to heat, bring to a high simmer, and cook, stirring occasionally, for 10 minutes. Taste and adjust, adding salt, pepper, and cayenne as desired. Continue to cook, stirring often, until thickened and reduced to about 2 cups.

Sauce will keep, refrigerated, for up to 4 days. Leftovers are excellent stirred into baked beans.

Chicken

Remove chicken pieces from brine and discard brine; place on a rack or platter and allow to come to room temperature, about 1 hour.

Preheat a grill to 375ºF.

Rub chicken pieces all over with olive oil and season with black pepper. Place on the grill, skin-side down, cover, and cook, undisturbed, for 8 minutes. Turn chicken, baste the skin generously with barbecue sauce, cover, and cook for another 8 minutes. Turn and baste; cook until skin is browned and sticky, 2 to 3 minutes. Turn and brown remaining side; if desired, repeat process once to further glaze chicken. It should register 160ºF at the thickest part. Remove to a platter and allow to rest for 5 minutes, then serve.