Melt butter and oil in a large soup pot over medium heat. Add onion, carrot, and celery, season with salt and pepper, then saute for 10 minutes or until vegetables are tender. Add garlic cloves then cook for 1 more minute, stirring constantly. Add chicken, season with more salt and pepper if chicken is unseasoned then stir to combine.
Add chicken broth, 1-3/4 cups milk, bay leaf, and parmesan cheese rind, if using, to the pot then turn the heat up to high to bring the soup a simmer. Add the rice then stir to combine. Place a lid on top then turn heat down to low and simmer for 10-15 minutes, stirring occasionally, until rice is tender.
In a small bowl whisk together remaining 1/4 cup milk with flour. Stream into the soup then stir to combine and cook uncovered for 5 more minutes. Turn off heat and remove pot from burner then add cheese and stir until smooth. Let soup sit and thicken for 10 minutes then serve.