Whisk together the buttermilk, lime zest and juice, garlic, chili powder, coriander, cumin, turmeric, hot sauce, cilantro, and salt and pepper, to taste, in a large bowl. Pour the marinade into a large resealable plastic bag, add the chicken, seal the bag and refrigerate. Marinate bone-in chicken up to 12 hours, boneless up to 4 hours, chicken legs or breasts, up to 24 hours. If using fish, marinate white fish such as snapper, halibut or bass up to 30 minutes.
Preheat the grill pan over medium-low heat for boneless breasts and fish, and medium heat for bone-in breasts, thighs and legs. Remove the meat/fish from the marinade and pat dry with paper towels. Brush with oil and season with salt and pepper, to taste, on both sides. Grill until golden brown and charred on both sides and just cooked through.