Remove the lower 2 inches of the lemongrass and the leaves, and chop the stems coarsely. Add lemongrass, ginger, garlic, shallots, cilantro, sugar, coconut milk, fish sauce, and soy sauce to the food processor bowl. Process until smooth.
Trim the chicken breast and put them in the gallon plastic bag (or marinating dish). Pour the marinade over the chicken, making sure the marinade coats the chicken completely. Leave in the refrigerator for 4 to 12 hours.
Preheat the gas grill or barbecue to a temperature of 400° F/200° C. Drain the chicken and reserve the marinade. Rub the hot grill with vegetable oil to keep the chicken from sticking.
Grill until the chicken is done, about 5 to 6 minutes per side. Brush with the reserved marinade while cooking.
Transfer cooked chicken to a serving platter, garnish with lime, cilantro, and chiles, and serve.
If you can't find lemongrass, use the zest of 1 lemon.
If you can't find palm sugar, use light brown sugar.