If using chicken breasts, pound the thicker parts to an even thickness with the flat side of a meat mallet; thighs can be left as is.
Place the chicken in a gallon-size resealable bag.
Combine green onions, jalapeño, ginger, garlic, soy sauce, lime juice, 2 Tbsp olive oil, brown sugar, thyme, allspice, black pepper, cinnamon, and nutmeg in a food processor and pulse several times until the mixture is somewhat coarse.
Pour the marinade into the bag with the chicken, rub to coat thoroughly, seal the bag, and refrigerate for at least 4 hours and up to 8 hours.
Preheat a grill to medium-high (about 400°F). Brush the grates with oil.
Remove chicken from the marinade and place on the grill. Grill until the internal temperature reaches 165°F, about 4–5 minutes per side depending on thickness.
Transfer the chicken to a plate, cover loosely, and let rest 5 minutes.
Serve the chicken warm over coconut rice and top with mango avocado salsa.