Cut the chicken into 1-1/2" pieces and set aside.
Make the marinade by combining the soy sauce, dark brown sugar, lime zest, vegetable oil, garlic, curry powder, ginger, cardamom and salt in a large bowl. Add the chicken pieces to the bowl and mix until chicken is evenly coated. Cover the bowl with plastic wrap and marinate in the refrigerator for at least 4 hours or overnight.
Make the sauce by whisking together the coconut milk, peanut butter, brown sugar, soy sauce and red curry paste in a medium saucepan. Bring to a simmer, then cook over medium heat, stirring occasionally, for about 3 minutes. Add the fresh lime juice. Set aside.
Thread the chicken onto skewers and grill.
Serve with rice and cucumber salad with mint.