Melt 1 tablespoon butter in a large stockpot over medium heat. Add onion, garlic, carrots, mushrooms, celery, and poultry seasoning and sauté until the onion is translucent, about 5 minutes. Add artichoke hearts and chicken broth and bring to a boil over high heat. Add rice and chicken, reduce heat, and simmer for 50-60 minutes or until the rice is tender. When the rice is tender, remove from heat and add salt and pepper.

When the soup has about 10 minutes left, melt the remaining 4 tablespoons of butter in a small saucepan over medium-low heat. Whisk in the flour to make a roux. Slowly add the milk, about 1/2 cup at a time, whisking constantly to remove any lumps, until smooth and thickened to the consistency of a thin pudding. Slowly whisk the milk mixture into the soup and stir until combined. Simmer 4-5 minutes or until slightly thickened and serve.