Start by melting the butter in a large pot or dutch oven over medium heat. Add the veggies and sauté until tender about 5-7 minutes. Grate the garlic into the pot and cook for 1 minute.
Add the chicken broth, rice and thyme. Taste and add salt and pepper as needed. Bring mixture to a boil, reduce heat, cover and simmer about 45 minutes until rice is tender.
During the last few minutes of cooking time prepare the roux. Melt the butter in a small saucepan over medium heat. Whisk in the flour and allow to cook for about a minute. Whisk in the half and half and bring mixture to a simmer. Allow to cook until slightly reduced and thickened.
Slowly add the roux to the soup mixture and stir together. Stir in the chicken and adjust seasoning to taste. Cook gently until the chicken is heated through.