Trim any visible fat from the chicken breasts and cut each breast in half lengthwise to create 8 even pieces.
Season both sides of the chicken with fresh ground black pepper.
Heat about 2 tablespoons peanut oil in a large heavy frying pan over medium heat until shimmering.
Add chicken pieces in a single layer and cook until nicely browned, about 5 minutes on the first side, then flip and cook 4–5 minutes more until an instant-read thermometer registers 165°F (75°C).
Remove cooked chicken to a plate and tent with foil to keep warm while you make the sauce.
Meanwhile, combine minced ginger, minced garlic, peanut butter, rice vinegar, soy sauce, Monkfruit sweetener, and red Thai curry paste in a food processor or blender and process until smooth; taste and add more curry paste if desired.
Reduce the pan heat to low and pour in the coconut milk, scraping up any browned bits from the bottom of the pan.
Stir the peanut butter mixture into the coconut milk and simmer gently for 2–3 minutes until heated through and slightly thickened.
Add any juices from the resting chicken into the sauce, simmer another minute, then turn off the heat and stir in most of the chopped cilantro, reserving a bit for garnish.
Serve the chicken hot with the peanut sauce spooned over and garnish with remaining cilantro if desired.