Put all ingredients EXCEPT OIL into a food processor or blender and blend until well combined.
With the blender running slowly drizzle in the oil then blend until completely smooth, about 30-60 seconds.
CHICKEN:
Marinate the chicken breasts in 1/2 cup of the Asian peanut sauce for about 20 minutes.
Remove chicken from marinade and cook in a non-stick skillet over medium heat until cooked through, about 5-8 minutes per side. Discard any remaining marinade.
Transfer chicken to a cutting board and let rest for 10 minutes.
Slice the chicken and garnish with cilantro and peanuts. Serve with the peanut sauce on the side.