Heat olive oil in a large pot over medium heat. Add onion, carrots, and celery; cook for 5 minutes until softened. Stir in garlic and thyme, cooking 1 minute more.

Pour in chicken broth, add bay leaf, and season with a pinch of salt and pepper. Bring to a boil, then add whole chicken breasts. Reduce heat to low, cover, and simmer for 15 minutes, or until chicken reaches an internal temperature of 165°F (74°C).

Remove chicken to a plate and let cool slightly. Use two forks to shred into bite-sized pieces. Discard bay leaf.

Return broth to a boil. Stir in rinsed rice, reduce heat, and cover. Simmer for 12-15 minutes until rice is tender.

Return shredded chicken to the pot. Stir to combine, and simmer 2-3 minutes to heat through. Taste and adjust salt and pepper.