1. Heat a large Dutch oven over medium-high heat. Add 1 1/2 teaspoons oil to pan; swirl to coat. Add half of chicken; cook 5 minutes, browning on all sides. Remove chicken from pan. Repeat procedure with remaining half of chicken. Remove from pan; discard drippings.
2. Place 1 cup broth and 1/2 cup corn in a food processor; process until corn is pureed.
3. Heat Dutch oven over medium-high heat. Add remaining 1 1/2 teaspoons oil to pan; swirl to coat. Add onion and carrot; sauté 2 minutes. Stir in pureed corn mixture, remaining 2 1/2 cups broth, remaining 1 cup corn, 1 1/2 cups water, oregano, and thyme; bring to a simmer. Stir in baking potato. Return chicken thighs to pan; cover, reduce heat, and simmer 20 minutes, stirring frequently to keep potato from sticking to pan. Add Yukon gold potato; cover and simmer 25 minutes, stirring frequently. Remove chicken from pan; cool slightly.
4. Remove chicken from bones, discarding bones. Shred chicken into bite-sized pieces. Return shredded chicken to pan. Stir in cilantro and next 4 ingredients (through black pepper); cook, uncovered, 5 minutes, stirring frequently. Ladle soup into bowls; top with avocado and capers.