Stir together brown sugar, soy sauce, sesame oil, fish sauce, chile garlic paste, and garlic in a large bowl. Add chicken and toss to coat. Remove chicken from marinade and thread onto skewers ( I ended up using 4 skewers total).
Transfer the leftover marinade to a small saucepan, and bring to a boil. Reduce heat and simmer until reduced by half, about 7–10 minutes.
Grill the chicken skewers, turning and basting often with reduced marinade, until cooked through, about 8–10 minutes.
Remove chicken pieces from skewers and serve atop crispy rice with sliced cucumber, mint, sliced scallion, chiles, chopped peanuts, and lime wedges.
*I found out the secret to making crispy rice, and I am never going back. Combine 1 cup of rice, 2 cups of water, and a pinch of salt in a heavy bottom pot with a lid. Bring to a boil, then reduce heat to low. Wrap the lid tightly in a clean kitchen towel, and place over pot to form a tight seal. Check rice around 30-35 minutes - it should be crispy and starting to turn golden on the bottom, but not burned. If you want it crispier, replace lid and continue to cook until bottom of rice reaches desired level of crispiness. When ready to serve, scrape out of pot and divide between bowls.