In a large saucepan, bring chicken broth and water to a boil and add the chicken. Add thyme and season with salt and pepper; add rice and reduce heat.
In a skillet, heat olive oil over medium-high heat. Add garlic, onion, carrots, and celery; cook until onions are translucent. Add flour to the mix and combine well. Add this mixture to the chicken broth mixture in the saucepan.
Stir milk into everything and continue to simmer, stirring occasionally until thickened, about 20-30 minutes.