In a 6- to 8-quart Dutch oven heat 1 tablespoon of the oil over medium heat. Add chicken; cook 6 to 8 minutes or until lightly browned, turning once. Remove chicken and set aside. Add carrots, sweet pepper, onion, and 1 tablespoon additional oil to the skillet. Cook 5 to 7 minutes or until vegetables begin to brown, stirring occasionally. Add broth, potatoes, and quinoa. Return chicken to pan. Bring to boiling; reduce heat. Simmer, covered, 10 minutes.
Meanwhile, husk and clean fresh sweet corn, if using. Cut kernels off 1 of the ears. Cut remaining ears crosswise into 2-inch slices.
Check chicken for doneness. If no longer pink (170 degrees F), transfer chicken to a cutting board. If not, leave in the pan. Add sliced ears of corn (or 1 cup thawed frozen corn) to the pan. Return to boiling; reduce heat. Simmer, covered, 5 minutes more or until potatoes and corn are tender and chicken is no longer pink (if still in the pan).
Coarsely shred chicken using two forks. Add shredded chicken and the Cilantro Oil to the soup. Cook 2 to 3 minutes over low heat to heat through.
In a medium skillet heat the remaining 1 tablespoon oil over medium-high heat. Add reserved corn kernels or remaining 1/2 cup thawed frozen corn. Cook 3 to 4 minutes or until corn is tender and golden brown, stirring occasionally.
Serve soup topped with corn kernels and lime wedges.
In a food processor or blender combine cilantro, olive oil, lime juice, garlic, crushed red pepper, and salt. Cover and process or blend until smooth, scraping sides of bowl as needed.