In a large Dutch oven, melt butter over medium heat.
Add onions, carrots, and celery; sauté for 7 minutes until softened.
Stir in minced garlic and cook for 1 minute until fragrant.
Pour in chicken stock and add chicken breasts, thyme, salt, and pepper.
Bring to a boil, then reduce heat to low and simmer for 20 minutes.
Remove chicken from the pot, shred into bite-sized pieces with two forks, and set aside.
Add the uncooked rice to the simmering broth and cook for 15 minutes or until rice is tender.
Return the shredded chicken to the pot and stir in the heavy cream.
Heat for 3 minutes until warmed through, ensuring the soup does not reach a hard boil.
Garnish with fresh parsley and adjust seasoning with additional salt or pepper if necessary.