Heat the olive oil in a large pot over medium-high heat. Add the diced onion, carrots, and celery, and sauté for 5-6 minutes until soft. Season with half of the kosher salt, pepper, and Italian seasoning.

Stir in the minced garlic and cook for an additional minute. Sprinkle the flour evenly over the vegetables, stirring until no white streaks remain.

Pour in the chicken broth, add the chicken breasts, rice, and bay leaf, along with the thyme and remaining seasonings. Stir well.

Bring to a boil, then reduce to medium-low and partially cover. Simmer for about 15 minutes until the rice and chicken are cooked through.

Remove the chicken and shred it into bite-sized pieces. Discard the bay leaf and any thyme stems.

Return shredded chicken to the pot, add the half and half, and stir to combine. Adjust thickness with more broth if needed and serve.