Heat the olive oil in a skillet over medium heat. Add the sliced sausage and cook for 3 to 4 minutes until lightly browned on both sides. Transfer the sausage to the slow cooker.
Add the diced onion, minced garlic, and carrots to the same skillet. Cook for 2 minutes, stirring often, then transfer everything to the slow cooker.
Pour in the chicken broth. Add the drained white beans, smoked paprika, salt, and black pepper. Stir to combine.
Cover and cook on low for 6 to 7 hours.
Stir in the chopped kale during the last 30 minutes of cooking so it softens without becoming mushy. Taste and adjust salt if needed before serving.