Preheat a grill to medium high and prepare for indirect grilling: On a charcoal grill, bank the coals to one side. On a gas grill, turn off half the burners. Toss the chicken with the ginger, 1 tablespoon fish sauce and 1 teaspoon each brown sugar, lime zest, lime juice and chili-garlic sauce. Season with salt and pepper. Place the chicken skin-side up on the cooler side of the grill; cover and cook until a thermometer inserted into the thickest part registers 165 degrees F, about 20 minutes. Move to direct heat and cook, uncovered, flipping occasionally, until the skin is charred and crisp, about 5 more minutes.
Meanwhile, make the dressing: Whisk 2 tablespoons water and the remaining 1 tablespoon fish sauce, 2 teaspoons brown sugar, 1 teaspoon lime zest, 2 teaspoons lime juice and 1/2 teaspoon chili-garlic sauce in a large bowl. Whisk in the vegetable oil.
Add the nectarines, mint, peanuts and 1/4 teaspoon salt to the bowl with the dressing; toss. Divide the chicken and nectarine salad among plates.