Heat oil in large heavy pot over medium-high heat. Add sausage; cook, stirring to crumble, until browned, 6 to 8 minutes. Transfer the sausage to a clean plate. Add onion, celery, and garlic to the pot; cook, stirring often, until softened, 3 to 4 minutes. Return the sausage and any accumulated juices to the pan.
Stir in 1 cup broth, scraping the bottom of the pot to loosen browned bits. Stir in sweet potatoes, tomatoes, the remaining 4 cups broth, pepper, and salt. If using fresh rosemary, tie the sprigs together with butcher's twine or secure in a cheesecloth bag. Add the rosemary bundle to the stew, gently pressing into the liquid (or sprinkle in the dried rosemary). Bring to a boil. Reduce to heat to low; simmer, uncovered, until the sweet potatoes are fork-tender, 15 to 20 minutes. Discard the rosemary sprigs, if using.
Stir in kale; cook until wilted, 1 to 2 minutes. Top each serving with Parmesan.
You can refrigerate the soup for up to 4 days or freeze for up to 4 months.