Combine curry paste and oil in a bowl. Stir until paste is softened.
Mix in 3/4 of the coconut milk and taste the mixture. If it tastes bland, add more curry paste and salt, until it is pleasantly pungent. If too spicy, add a little more coconut milk.
Place chicken into a plastic zip lock bag, pour in marinade and seal. Toss to coat evenly. Allow to sit at room temperature for 20 minutes. For a more intense flavor, marinate for several hours in refrigerator.
Drain and discard the marinade. In a skillet or on a grill, on medium heat, grill the chicken 10 to 12 minutes, turning onto each side twice. Brush each side with a few tablespoons of chili sauce.
The chicken is done when it is white throughout, but still juicy and the sauce is a bit caramelized in spots.
Serve the chicken breasts whole or thickly sliced and mounded on a platter over rice. Sprinkle with chopped peanuts and chopped cilantro. You can also serve a little chili sauce on the side.