Discard skin from chicken. Remove meat from bones; cut into strips. In a medium saucepan bring water to boiling. Break each package of noodles into 4 pieces (reserve seasoning packets for another use). Add noodles to boiling water. Remove from heat; cover and let stand for 5 minutes. Drain noodles. Rinse noodles with cold water; drain. Set aside.
Meanwhile, in a large bowl whisk together the peanut butter, coconut milk, lime juice, cilantro, sugar, salt, and cayenne pepper until smooth. Add chicken, noodles, green onions, cucumber, and, if desired, sweet pepper; toss to coat.
Serve immediately or cover and chill for up to 2 hours. If desired, garnish with peanuts.