Preheat a grill for cooking over medium heat. Whisk together the ginger, 2 tablespoons coconut milk, 1/4 cup fish sauce, 2 tablespoons honey, juice of 1 lime, half the minced garlic, 1 teaspoon salt and a few grinds of pepper in a large bowl. Pierce the skin of the drumsticks all over with the tip of a knife or fork. Toss the drumsticks with the marinade to coat and set aside.
Meanwhile, heat the canola oil in a small saucepan over medium heat and add the remaining garlic. Cook, stirring often, until the garlic is golden, about 30 seconds. Remove from the heat and whisk in the remaining 1/2 cup coconut milk, 1 teaspoon fish sauce, 1 tablespoon honey and juice of 2 limes. Whisk in the peanut butter and 1 tablespoon water. Stir until well combined and season with 1/4 teaspoon salt. Set aside.
Grill the drumsticks, turning halfway through and basting twice with the marinade, until they are charred and crisp, 8 minutes. (Discard the remaining marinade.) Reduce the heat to low, then cover and cook until the drumsticks are tender, crisp and cooked through, 25 to 30 minutes. Remove the drumsticks to a bowl. Add half of the peanut sauce and toss to coat. Sprinkle with the cilantro leaves and serve with the lime wedges and remaining peanut sauce on the side.
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