Set a large pot or Dutch oven over medium heat. Add the olive oil and onions and cook, stirring often, until the onions are softened and translucent.
Add the chopped carrots and celery, minced garlic, bay leaves, Italian seasoning, salt, and black pepper. Mix well to combine with the onions. Cook, stirring often, for 6-8 minutes, or until the veggies start to soften and the herbs turn fragrant.
Mix in the chicken broth plus six cups of water. Bring it to a simmer over high heat.
Once simmering, reduce heat to medium and add the chicken breasts to the pot. Simmer the chicken breasts just until they are fully cooked - about 18-20 minutes. Then remove the chicken from the broth and set it aside on a cutting board or plate.
Add the noodles to the soup, mixing well to prevent the noodles from clumping (see notes).
In a small bowl, whisk together the half-and-half and flour until smooth, then whisk it into the soup. Return to a simmer (do not boil) and cook for 8-10, stirring frequently.
Finally, shred the cooled chicken into small pieces and return the cooked chicken back to the pot. Season with additional salt and black pepper to taste.