Mix together the coconut milk, peanut butter, sugar, Thai red curry paste, 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.
While chicken is marinating, prepare the satay sauce. Mix together the ingredients in a large bowl or liquid measuring cup and set aside.
Heat oil in large oven-proof skillet over medium-high heat. Cook the chicken about 4 minutes per side or until cooked through. Add the sauce and simmer for 5 minutes. Top with some fresh cilantro and additional lime juice if desired.
Notes
If short on time, just skip the marinade with the chicken.
Control the heat by adjusting the amount of Thai red curry paste. For milder tastes, start with a smaller amount and add more gradually, tasting as you go.
To add some crunch and little pieces of whole peanuts throughout the sauce, try crunchy peanut butter.
Regularly taste the curry as it simmers. Adjust the seasonings such as salt, soy sauce or lime juice to achieve a well-balanced dish.
Leftovers will store well in the fridge for 2-3 days in an air-tight container or be frozen for 3 months.