Sticky Guava Barbecue Drumsticks

Mix together 1 Tbsp. achiote (annatto) powder or paprika, 1 Tbsp. Diamond Crystal or 1¾ tsp. Morton kosher salt, 1 Tbsp. garlic powder, 1 Tbsp. ground coriander, 1 Tbsp. ground cumin, 2 tsp. dried oregano, and 2 tsp. freshly ground black pepper in a small bowl. Generously sprinkle seasoning all over 8–12 large chicken drumsticks or 4 thighs and 4 drumsticks (about 3 lb.). Let sit while you make the barbecue sauce. (If you have time, arrange chicken on a wire rack set inside a rimmed baking sheet and let chill, uncovered, overnight.)

Assemble your ingredients: Chop 8-oz. guava or quince paste into ½" cubes (if using 8 oz. guava jelly, just add directly below, no need to prep). Peel and finely chop one 2" piece ginger and 2 garlic cloves. Finely chop 1 canned chipotle chile in adobo sauce.

Bring guava paste, ginger, garlic, chipotle chile, ½ cup apple cider vinegar, ⅓ cup spiced or dark rum, 3 Tbsp. fresh lime juice, 3 Tbsp. ketchup, 1 Tbsp. soy sauce, ½ tsp. ground allspice and ¼ cup water to a simmer in a medium pot over medium-high heat, about 5 minutes. Reduce heat to medium and continue simmering, stirring occasionally, until guava melts and sauce is thick but pourable, about 10 minutes. Scoop out ½ cup sauce and set aside for serving.

If grilling, prepare a grill for medium-high heat. Grill chicken, turning occasionally, until lightly charred, 15–20 minutes. Baste chicken with guava sauce and continue grilling, turning and basting often (try to use most, if not all, of the sauce) and moving to a cooler part of the grill if you have flare-ups, until cooked through and an instant-read thermometer registers at least 165° (we like the extra tenderness of bringing it to 175°, so feel free to go a bit further), 8–10 minutes longer.

If baking, place a rack in very top position of oven and remaining rack in the middle. Heat broiler. Line a large baking sheet with foil and set a wire rack inside. Arrange chicken on rack and broil, undisturbed, until charred in spots, 5–7 minutes. Remove chicken from oven and turn over. Broil until other side is charred in spots side, 3–5 minutes. Brush chicken with guava sauce. Reduce oven temperature to 350° and move chicken to middle rack. Bake 10 minutes. Turn over and brush with more sauce. Continue cooking, basting with more sauce every few minutes (try to use most, if not all, of the sauce), until an instant-read thermometer registers 165° (we like the extra tenderness of bringing it to 175°, so feel free to go a bit further), 20–25 minutes longer.

Transfer chicken to a platter and let rest 5–10 minutes. Serve with reserved guava sauce for dipping.

Do ahead: Sauce can be made 1 week ahead. Cover and chill.