Heat the olive oil in a large deep skillet or Dutch oven over medium-high heat. Add the sausage and cook, breaking it into bite-size pieces, until browned, about 5 minutes. Spoon off excess fat, leaving about 1 tablespoon.
Add the onion and cook until softened, about 4 minutes. Stir in the garlic and cook 30 seconds, until fragrant.
Add the tomatoes with their juices, broth, beans, Italian seasoning, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Bring to a simmer, then lower the heat and cook 10 minutes, stirring occasionally, until slightly thickened.
Add the kale a handful at a time, stirring until it wilts, 3 to 4 minutes total.
Stir in the Parmesan, taste, and adjust salt and pepper. Serve hot with extra Parmesan and crusty bread.