Put the cilantro, ginger, shallot, and garlic in the bowl of a food processor; pulse to chop. Add the Cayenne Pepper Sauce or hot sauce, olive oil, lime juice, salt, and pepper; process until fine.

Pour half the marinade in a non-reactive baking dish. Add the chicken pieces and turn to coat. Pour the remaining marinade over the chicken, cover, and refrigerate overnight and up to 3 days.

On a covered grill over indirect heat, arrange chicken skin-side up and cook until an instant-read thermometer inserted into the thickest piece of chicken reads 165F and the skin is browned. Remove to a platter and pour glaze (method below) over chicken.

Meanwhile, make the glaze: Melt the butter in a saucepan over medium-high heat. Add the garlic and cook until the garlic just begins to brown; do not burn. Add the cilantro and stir until wilted. Reduce the heat to low, add the Cayenne Pepper Sauce or hot sauce and lime juice; simmer 5 minutes. Use at once, or cover and refrigerate.