Step 1

Heat olive oil in a small saucepan over medium. Cook garlic, turning occasionally, until golden brown, about 4 minutes. Add tomato paste and cook, stirring constantly and scraping bottom of pan, until slightly darkened, about 3 minutes. Add maple syrup, soy sauce, Worcestershire sauce, vinegar, and Sriracha. Bring to a boil, then reduce heat to medium-low and simmer until flavors have come together, about 1 minute. Remove from heat. Transfer about half of barbecue sauce to a heatproof bowl and set aside for serving.

Step 2

Prepare a grill for medium-high indirect heat (for a charcoal grill, bank coals on one side of grill; for a gas grill, leave one or two burners off); oil grate. Season chicken generously with salt. Grill chicken over direct heat, turning every minute, until browned on all sides, about 5 minutes. Move chicken over indirect heat. Cover, positioning vent (if your grill has one) over chicken, and continue to grill, keeping grill covered as much as possible and turning every 5 minutes or so, until an instant-read thermometer inserted into thickest part registers 140°F–145°F, 18–25 minutes depending on the size of the pieces. Baste chicken with barbecue sauce and grill, turning and basting occasionally, until an instant-read thermometer inserted into thickest part registers 160°F for breast and 165°F for dark meat, about 10 minutes longer.

Step 3

Arrange chicken on a platter. Serve with reserved barbecue sauce alongside.

Step 4

Do Ahead: Barbecue sauce can be made 2 weeks ahead. Transfer to an airtight container; cover and chill.