Juice limes (1/2 cup); chop shallots. Peel and grate ginger; mince garlic. Combine in zip-top bag: 1/2 cup soy sauce, 1/4 cup lime juice, brown sugar, shallots, garlic, and 2 teaspoons ginger. Cut chicken into 1-inch-thick strips and add to bag (wash hands). Seal bag and knead to coat; marinate 2 hours.
Preheat grill on medium-high. Remove chicken from zip-top bag (discard marinade). Thread chicken onto individual skewers; grill skewers 1–2 minutes on each side until chicken is grill-marked and 165°F.
Whisk in small bowl: peanut butter, coconut milk, honey, sriracha, remaining 1/4 cup lime juice, remaining 2 tablespoons soy sauce, and remaining 2 teaspoons ginger until blended. Chop peanuts. Place skewers on a serving plate and sprinkle with chopped peanuts and serve with peanut sauce on the side.