In a large Dutch oven or stock pot, combine liquids, vegetables, and seasonings.
Simmer over medium-high heat 10 minutes to extract vegetables' favors. Don't cook them until they turn mushy--it's nice having some texture in a cheese soup.
Make roux in a saucepan by melting butter over medium-low heat, then stirring in flour.
Cook 2 minutes, stirring constantly, then whisk in warm milk. Stir and gently simmer another minute to eliminate any "floury"flavor.
Increase heat under stock and bring to a boil. Add roux-thickened milk to the stock and whisk vigoursly to prevent lumping and scorching. The stock will be dark, but will continue to get lighter when the cheese is added.
Bring the soup to a simmer, then turn off the heat and add the cheese all at once. Again, whisk vigorously until the soup is creamy and smooth.
To serve, ladle into bowls and top with sausage garnish.