Heat oil in a large Dutch oven over medium-high heat. When hot, add the chicken sausage then cook before breaking up with a spatula. Continue cooking until the sausage is fully cooked. Use a slotted spoon to transfer the sausage to a bowl.
If the pot is dry, and another tablespoon or two of olive oil. Add the diced mushroom caps to the pot then cook before adding the diced onion, carrots, and celery. Season with kosher salt and black pepper. Lower the heat to medium, then cook, stirring occasionally, until the vegetables are soft.
Stir in wild rice, then add chicken stock. Return the cooked sausage to the pot along with a Parmesan rind (optional) and sprigs of fresh rosemary. Cover with a tight lid, then cook over medium heat. While the soup is cooking, make the thickening slurry by whisking cornstarch and water in a small bowl.
Carefully lift the lid, then remove the Parmesan rind and herb bouquet. Increase the heat to medium-high, then pour in the cornstarch slurry. Add chopped kale (or spinach) and stir well. Bring the mixture to a boil.
Once the soup is thick and creamy, turn off the heat, then pour in half-and-half. To serve, ladle into bowls then top each portion with a drizzle of olive oil, a sprinkle of grated Parmesan, and extra black pepper. Enjoy!