Cook the veggies. Heat the butter in a large pot or Dutch oven over medium heat. Add the onions and cook, stirring often, for 2-3 minutes, or until translucent. Add the carrots and celery, mix thoroughly, and continue to cook for 10-15 minutes (note: if the vegetables start to brown or burn, reduce heat to medium-low).
In the last minute, stir in the fresh minced garlic cloves, fresh thyme, and season with salt and pepper.
Add the flour. Sprinkle the flour over the vegetables and mix thoroughly until fully incorporated with the veggies. Cook, stirring continuously for 3-5 minutes, or until the flour is lightly toasted (do not burn the flour).
Add the rice and thoroughly mix to combine with the flour-coated veggies.
Stirring continuously, slowly add the stock and water to the pot. Mix thoroughly to prevent any clumping. Add the bay leaves and bring to boil.
Cook. Once boiling, reduce heat to medium-low. Cover, half-way, with a lid and simmer for approximately 30-45 minutes, stirring occasionally. Add the chicken breasts and cover. Allow chicken breasts to cook for approximately 25-30 minutes (maximum). Use kitchen tongs to remove the chicken breasts from the pot and set aside to cool.
Shred the chicken and return to the pot. Once the chicken is cool enough to handle, use two forks or your hands to shred into small pieces. Return the shredded chicken to the pot. Increase heat to medium and allow soup to simmer for 5-10 minutes. Season with additional salt and pepper, to taste.
Add the heavy cream. Remove the soup from heat and stir in as much (or as little) heavy whipping cream as you'd like (I added approximately 3/4 cup).
Serve. Serve garnished with fresh chopped parsley or fresh thyme, if desired.