Heat 1-2 tablespoons of olive oil in a Dutch oven or heavy bottom stock pot and add seasoned chicken thighs. Brown well on both sides, about 6-7 minutes on each side. Transfer chicken thighs to a separate plate.
Adjust the heat to medium and if needed, add another tablespoon of olive oil to the Dutch oven. Add the carrots, celery, and onions. Sauté for a couple minutes and add the garlic. Cook for another minute.
Add the broth and bay leaves to the pot, increase the heat to high, and bring to a boil. Once boiling, add the rice and reduce heat to medium-low. Simmer for 20 minutes. Shred the cooked chicken thighs while the soup is simmering.
When the rice is tender, add the shredded chicken to the soup and let simmer while you prepare the avgolemono sauce.