Cut the chilies into thin rings. Remove one-third of the chilies and set aside for garnish. Place the remaining chilies, garlic, and sugar in a mortar and pound into a coarse, wet paste. (If you don’t have a mortar, just chop with a knife.) Transfer to a small bowl and add the water, lime juice and fish sauce. Stir well to dissolve. Add the reserved chilies and carrots. Set aside for 10 minutes before serving.
Note: Nuoc cham is very amenable to variations and adaptations. In Vietnam, cooks like to use various vegetables to flavor the sauce, such as thinly sliced marinated daikon and carrots, ginger, scallion oil or peanuts and even slices of kohlrabi and the core of a white cabbage. Each imparts a distinctive savoriness.