1. Rinse chicken and pat dry; place in a bowl. In another bowl, combine lime juice, olive oil, sesame oil, soy sauce, sugar, garlic, and ginger. Pour about 1/4 cup of the mixture over chicken, reserving remainder. Chill chicken for 30 minutes.
2. Lift chicken from marinade (discard used marinade) and lay over a solid bed of medium coals or medium heat on a gas grill (you can hold your hand at grill level only 4 to 5 seconds); close lid on gas grill. Cook, turning chicken as needed to brown evenly, until no longer pink in the center of the thickest part (cut to test), 6 to 8 minutes total. Transfer to a board and let stand until cool enough to handle, then cut across the grain into 1-inch-wide slices.
3. Meanwhile, in a covered 6- to 8-quart pan over high heat, bring about 3 quarts water to a boil. Add the pasta and cook, stirring occasionally, until tender to bite, 4 to 6 minutes. Drain pasta and rinse under cold running water until cool.
4. In a large bowl, combine pasta, reserved lime-juice mixture, chicken, bell peppers, cucumber, parsley, green onions, and chili flakes; mix to coat. Serve immediately or cover and chill up to 1 day.