Heat the olive oil in a large pot over medium heat. Add the diced yellow onion and minced garlic. Sauté until tender, about 5 minutes.
While the onion and garlic sauté, clean and slice the carrots and celery.
Add the sliced carrots and celery to the pot with the onions and garlic. Cook for about 5 minutes more, stirring occasionally.
Add the whole chicken breast, bay leaf, dried oregano, dried thyme, cracked black pepper, 1/2 cup dry rice, and 6 cups of water to the pot. Do not add salt yet.
Bring the pot to a boil over medium-high heat, then reduce the heat to low and let simmer for one hour, or until the chicken is cooked through and the rice is tender. Remove the bay leaf.
Carefully remove the chicken breast from the pot and shred it with two forks. Return the shredded chicken to the soup. Stir in salt to taste.