Finely grate the ginger into a fine sieve set over a bowl. Press the juice from the ginger; you should have 1/3 cup. Discard the solids. Whisk in the honey and a generous pinch each of salt and pepper.
Light a grill and oil the grate. In a large bowl, toss the chicken with the 2 tablespoons of oil and season generously with salt and pepper. Grill over moderate heat, turning, until lightly browned on both sides, about 7 minutes. Continue to grill, turning and brushing with the ginger-honey mixture occasionally, until lightly charred and cooked through, about 8 minutes longer. Transfer to another large bowl and toss with the peanuts and cilantro. Serve with lime wedges.