Dissolve 1/4 cup kosher salt in 2 quarts of cold water in a large bowl.

Add the chicken pieces to the brine and refrigerate for 2 hours.

Melt butter in a medium saucepan over low heat.

Add minced onion and garlic to the saucepan and cook until softened.

Stir in paprika, ancho chile powder, and crushed red pepper; cook for 1 minute.

Add 1 cup water, brown sugar, and cider vinegar to the saucepan; bring to a simmer.

Stir in tomato paste and molasses; simmer over low heat, stirring occasionally, until the sauce thickens (about 10 minutes).

Season the barbecue sauce with salt to taste; let it cool.

Preheat grill to medium-low or oven to 500°F.

Remove chicken from brine and pat dry.

Grill chicken over moderately low heat, turning occasionally, until cooked through (25 minutes for white meat, 35 minutes for dark meat).

Alternatively, roast chicken skin side up on a rimmed baking sheet in the upper third of the oven until cooked through (25 minutes for white meat, 35-40 minutes for dark meat).

Set aside half of the barbecue sauce for serving.

Brush the remaining barbecue sauce onto the chicken during the last 10 minutes of grilling or roasting.

Continue grilling or roasting until the chicken is nicely glazed.

Serve the barbecued chicken with the reserved barbecue sauce.