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 soft. 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. Turn heat up to bring to a boil then add rice and stir to combine. Place a lid on top then turn heat down to medium-low and cook 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 soup, stir to combine, then 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.
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