Melt butter in a large soup pot over medium heat. Add shallot, carrots, celery, and seasonings and sauté for 4-5 minutes, or until shallots are tender and translucent.
Whisk in flour and cook for another 1-2 minutes, to "cook off" flour flavor. Stir frequently to avoid burning.
Slowly whisk in broth, in small increments, until a slurry forms. Once all broth has been added, bring to a boil and add rice and mushrooms. Reduce heat to medium and simmer for 45-50 minutes, or until rice is tender.
Remove from heat and stir in chicken and milk. Stir well to combine.