Heat barbecue to medium. Peel and halve pineapple into 6 ½-inch thick slices. Place chili pepper, 5 tsp of the sugar and 1 tsp of the paprika in mortar and pestle; grind until fine. Sprinkle over and press on pineapple slices and let marinate for 15 minutes.
Meanwhile, place chicken in a large resealable bag. Finely grind remaining 1 tsp sugar, salt, oregano, garlic, thyme and cayenne in mortar and pestle. Sprinkle mixture over chicken; seal bag and massage onto drumsticks. Let rest at room temperature for 10 minutes.
Remove drumsticks to a baking sheet and brush with all but 2 tsp of the oil. Grill chicken, covered on a greased grill, flipping every 8 minutes, until juices run clear when thickest part is pierced, about 20 minutes.
Brush pineapple with remaining 2 tsp oil. Grill until caramelized, about 2 minutes. Serve drumsticks with pineapple and torn cilantro.