Mix together the coconut milk, peanut butter, sugar, soy sauce and fish sauce in a large shallow bowl until combined and add chicken, turning to coat completely. Put chicken in fridge for 30 minutes.
Heat oil in large oven-proof skillet over medium-high heat. Cook chicken about 4 minutes per side or until cooked through. Add half of the peanut sauce so it is coated. Save remaining sauce for dipping if needed. Top with some fresh cilantro and additional lime juice if desired.
While chicken is cooking, combine all of the peanut sauce ingredients in a saucepan except the lime juice. Bring to boil and reduce heat, allowing to simmer until thickened. (About 5 minutes) Take sauce off heat and add lime juice. Adjust for salt if needed.
Use chicken cutlets or cut chicken breasts in half so the chicken is thin enough to cook quickly.
Add cold oil to a cold skillet and let heat up to proper temperature before adding the chicken. You’ll know it is ready when the oil starts to shimmer.
You know the chicken is cooked when it reaches 165°F with an instant-read thermometer or if you cut into the chicken and the juices run clear and there is no pink. Honestly, I always just cut the chicken to see when it is done. If it isn’t just cook slightly longer.
Cook the peanut sauce for at least 5 minutes or until it has thickened. You want a sauce with some body to it when you pour over the chicken.
Leftovers will store well in the fridge for 2-3 days in an air-tight container or be frozen for 3 months.