Add first nine ingredients, tomatoes through black pepper, to the bowl of a 4-quart slow cooker. Cover and cook on low for 5-7 hours, or until vegetables are softened.
About 30 minutes before serving the soup, prepare the roux. Heat oil and butter over low heat in a sauce pan and add flour. Stir constantly with a whisk for 5-7 minutes, until roux is a smooth light brown paste. Slowly stir 1 cup of hot soup into the roux, until fully mixed in. Add another 3 cups soup to the mixture, and continue to stir until smooth. Add thickened soup back into the slow cooker, and mix in the Parmesan and milk.
Cover and cook on low for another 30 minutes. Taste for seasonings, and enjoy.
*Use the milk of your choice: half and half will make a thicker richer soup that's higher in calories, skim milk will make a thinner soup that’s lower in calories. I like the compromise of whole milk in this soup. You could even use a non-dairy milk like canned coconut milk (that would give a great thick soup), or almond milk.