Heat oil in a large Dutch oven over medium-high. Add sausage; cook, stirring often and breaking into small crumbles using a wooden spoon, until sausage is rendered and browned, 5 to 8 minutes.
Add onion and celery to Dutch oven; cook, stirring often and scraping bottom of pot to loosen any browned bits, until vegetables begin to soften, 4 to 6 minutes. Stir in sun-dried tomatoes and garlic. Cook, stirring often, until fragrant, about 1 minute. Stir in stock and chickpeas; bring to a boil over medium-high. Reduce heat to medium-low; simmer, uncovered, until flavors meld, about 15 minutes. Spoon off and discard any foam that rises to the surface.
Add kale to Dutch oven; cook, stirring occasionally, until tender, about 5 minutes. Remove from heat, and stir in cheese, cream, basil, and salt. Divide soup evenly among bowls, and garnish with additional cheese, if desired.
