Prepare the Vegetables: Heat the olive oil in a large, heavy saucepan over medium heat. Add the onion, carrots, celery, and garlic. Cook until the vegetables are soft and the onion is translucent.
Cook the Chicken and Rice: Add the chicken, rice, bay leaves, 2 sprigs of thyme, and chicken broth. Bring to a boil, then reduce the heat and simmer, covered, for 20-25 minutes until the rice is cooked.
Add Milk: Pour in the heavy cream and warm through.
Finish and Serve: Remove from heat. Garnish with the remaining sprig of thyme. Serve with warm crusty bread and a large salad.