Melt butter in large stockpot over medium heat. Stir in the onion, celery, and carrots and saute until onions are translucent and carrots are tender. Add the flour and stir well. Gradually pour in the chicken broth, stirring constantly, until all has been added. Bring to a boil, reduce heat to low, and let simmer.
Add rice, chicken, salt, spices to taste, and white wine. Let simmer for 1 to 2 hours, then stir in milk and allow to heat through.