Microwave the onion, 1 tablespoon vegetable oil, and tomato paste in a microwave-safe bowl, stirring occasionally until the onion is softened (about 5 minutes). Transfer this mixture to the slow cooker.

Add chicken pieces (legs, backs, wings), water, soy sauce, salt, thyme, and bay leaves into the slow cooker. Cover and cook until the broth develops a rich flavor, 7 to 8 hours on low or 4 to 5 hours on high.

Use a slotted spoon to discard the chicken pieces. Strain the broth through a fine-mesh strainer into a large container, discarding solids. Let the broth settle for 5 minutes, then skim any fat off the surface. Measure to ensure you have 9 cups; add water if needed.

Microwave the carrots, celery, and 2 teaspoons of vegetable oil in a bowl, stirring occasionally, until softened (5 to 7 minutes). Return the strained broth to the slow cooker, and add the microwaved vegetables along with the chicken breasts. Cover and cook on low for 3 to 4 hours, or until the chicken is cooked through (internal temperature of 165°F) and vegetables are tender.

Remove chicken breasts from the slow cooker and transfer to a cutting board. Let cool slightly, then shred into bite-sized pieces using two forks. Stir the shredded chicken back into the soup and let sit until heated through (about 2 minutes).

Season the soup with salt and pepper to taste. Serve hot.