Heat the olive oil in a large pot or Dutch oven over medium heat. Add the diced onion, carrots, and celery, sautéing for 5-7 minutes until they begin to soften.
Stir in the minced garlic, dried thyme, salt, and black pepper. Cook for 1 minute until fragrant.
Pour in the chicken broth. Add the whole chicken breasts and the uncooked white rice, then bring the mixture to a boil.
Reduce the heat to low, cover the pot tightly, and simmer for 15 to 20 minutes, or until the chicken reaches an internal temperature of 165°F and the rice is tender.
Carefully remove the cooked chicken breasts from the pot. Use two forks to shred the meat, then return the shredded chicken to the soup.
Stir in the frozen peas and heavy cream. Allow the soup to simmer uncovered for 3-5 minutes until heated through and slightly thickened.
Remove the pot from the heat. Stir in the fresh chopped parsley and serve hot with crusty bread for dipping.