1. Rinse chicken and pat dry; trim off and discard excess fat. Put chicken in a large bowl. Stir piri-piri marinade, pour over chicken, and turn pieces to coat. Cover and chill at least 4 hours or up to 1 day, turning chicken occasionally.
2. With tongs, lift chicken from marinade, drain well, and lay on a barbecue grill over a solid bed of medium coals or medium heat on a gas grill (you can hold your hand at grill level only 4 to 5 seconds; keep a spray bottle of water handy to douse any flare-ups); close lid on gas grill. Cook, turning occasionally, until skin is well browned and meat at bone is no longer pink (cut to test), 35 to 40 minutes total; brush occasionally with marinade until about 10 minutes before chicken is done. As pieces are done, transfer to a platter and cover loosely with foil to keep warm.
3. Pour remaining marinade into a 1 1/2- to 2-quart pan over medium-low heat. Stirring occasionally, bring mixture to a simmer; adjust heat to maintain simmer. Add butter and stir until melted and incorporated; turn heat to low and stir occasionally until ready to serve. Pour sauce into a small bowl or pitcher.
4. Garnish chicken with lemon wedges and serve with sauce.