Combine garlic, chillies, sugar and 1/3 cup of lime juice in a small food processor. Process until well combined. Pour into a jug. Add fish sauce and water. Stir to combine.
Place the mushrooms and chicken in separate bowls. Pour 1/4 cup marinade over the mushrooms and 1/4 cup over the chicken. Cover and refrigerate for 30-60 minutes.
Thread the chicken, lime leaves, mushrooms and lime wedge alternately onto skewers.
Preheat barbecue plate on medium-high. Lightly grease the plate and cook skewers, basting with a little of the remaining marinade, 5-8 minutes, turning until cooked through. Serve with remaining marinade as dipping sauce.