In a slow cooker, combine all ingredients except for milk, cheese, and fresh basil. Cook on high for 4 hours or low for 8 hours.
After the time has completed mash the vegetables in the stew. For a creamier stew, blend the cooked vegetables in a blender and return to the slow cooker. Stir in the milk and cook, uncovered, on low for 30 minutes.
Ladle into bowls and top with parmesan and fresh basil. Serve and enjoy.