Step 1

Heat olive oil in a small saucepan over medium until hot. Cook garlic, turning occasionally, until golden brown all over, 4–5 minutes. Add tomato paste and cook, stirring constantly and scraping bottom of pot, until paste darkens, 3–4 minutes. Add maple syrup, soy sauce, Worcestershire sauce, vinegar, and Sriracha. Bring to a boil, then reduce heat to medium-low and simmer to let flavors meld, about 1 minute. Remove from heat. Transfer about half of barbecue sauce to a heatproof bowl; reserve 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 grates. Season chicken generously with salt. Grill chicken over direct heat, turning every minute, until browned on all sides, 5–6 minutes. Move chicken over indirect heat 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°–145°, 18–25 minutes depending on size. Begin basting with barbecue sauce, turning chicken occasionally, until an instant-read thermometer inserted into thickest part registers 160° for breast and 165° for dark meat, about 10 minutes longer.

Step 3

Transfer chicken to a platter. Serve with reserved sauce alongside.

Step 4

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