Heat the olive oil in a large pot or Dutch oven over medium heat. Add the Italian sausage and cook, breaking it apart with a wooden spoon, until browned and cooked through, about 5-7 minutes.
Add the diced onion and minced garlic to the pot. Sauté until the onion is translucent, about 3-4 minutes.
Stir in the diced sweet potatoes, ground cumin, and smoked paprika. Cook for 2-3 minutes, allowing the spices to bloom.
Pour in the vegetable or chicken broth and add the diced tomatoes (with their juices). Bring the mixture to a boil, then reduce the heat to a simmer.
Cover and let the soup simmer for about 20 minutes, or until the sweet potatoes are tender.
Add the chopped kale (or spinach) to the soup and cook for an additional 5 minutes, until the greens are wilted.
Season the soup with salt and pepper to taste.
Serve hot, garnished with fresh parsley if desired.