In a large resealable plastic bag or bowl, combine the curry paste, peanut butter, coconut milk, and salt. Add the chicken. Seal the bag (or cover the bowl with plastic wrap). Let marinate for at least 1 hour and up to 24 hours.
Preheat a gas grill to 400 F (205 C). Grates should be clean; scrape if needed.
Place the chicken to the hot grill. (No need to wipe off the marinade. Just keep whatever naturally comes up with each chicken breast.) Close the lid.
Grill until the internal temperature, inserted into the thickest part of the chicken, reaches 165 F (75 C). This should take about 15 minutes (6 to 8 minutes per side).
Transfer the chicken to a serving platter and cover with foil. Let rest 5 to 10 minutes. Sprinkle with peanuts and cilantro and serve with lime wedges.