Heat olive oil in a medium pot or Dutch oven over medium-high heat. Add sausage and break up with a wooden spoon until sausage is no longer pink. Transfer sausage to a bowl and set aside.
Heat remaining pan drippings over medium-high heat. Add onion, squash, and potatoes and stir 3-5 minutes until onions are translucent. Add garlic and flour and cook 1 additional minute until fragrant.
Add chicken broth to pot and bring to a simmer over medium-high heat. Reduce heat to low, cover pot, and allow to cook 10-15 minutes until squash and potatoes are tender, stirring occasionally.
Scoop 1-/2 cups of butternut squash and potatoes out of pot and puree in a blender or food processor until smooth.
Add pureed mixture back to pot with sausage, heavy cream, and kale and allow to simmer over medium heat 3-5 additional minutes until kale is tender.
Season soup with salt and pepper to taste. Serve soup with grated Parmesan on top with bread for dipping. Enjoy!