Preheat oven to 450 degrees F. Coat a large baking sheet with cooking spray.
Break dough into 8 equal pieces and roll each piece into a thick rope shape. Twist each 'rope' piece into a knot. Place balls on prepared baking sheet and spray with cooking spray. Sprinkle with Parmesan to coat surface. Bake 8 minutes, until golden brown.
Meanwhile, melt butter in a large stock pot over medium heat. Add mushrooms, onion and garlic and cook 5 minutes. Add thyme and cook 1 minute. Add chicken broth and wine and bring to a simmer, about 1 minute. Simmer 5 minutes.
While soup is simmering, blend softened butter with chopped fresh parsley and salt and pepper. Remove Parmesan knots from the oven.
Remove soup from the heat and, using an immersion or stand blender, puree until smooth Return the soup to low heat, add milk and andouille sausage and simmer 1 minute to heat through.
Ladle soup into bowls and top with arugula and walnuts. Serve rolls on the side with parsley butter.