Place the Italian sausage and diced onion in a large frying pan over medium-high heat. Cook for 4–5 minutes, breaking up the sausage as needed, until the meat is well browned and the onion has softened.
Stir the minced garlic into the sausage mixture and continue cooking for 1 additional minute, just until aromatic.
Transfer the cooked sausage, onion, and garlic mixture to a 4–6-quart slow cooker. Add the chicken broth, potato chunks, salt, black pepper, and a pinch of red pepper flakes. Stir until the ingredients are evenly distributed.
Cover and cook on Low for 4–5 hours or on High for approximately 2 hours. Slow cooker temperatures can vary by model, so continue cooking as necessary until the potatoes are tender.
Once the potatoes have softened, stir in the chopped kale, heavy cream, and some of the grated Parmesan. Mix thoroughly, allowing the residual heat to gently wilt the kale.
Taste the soup and adjust the seasoning as needed. Serve warm with the remaining freshly grated Parmesan as desired.