Heat the oil in a skillet over medium-high heat and brown the sausage, breaking it into bite-size crumbles, 7 to 8 minutes.
Add the onion and carrots and cook for 4 minutes, then stir in the garlic for 30 seconds.
Transfer the mixture to a 6-quart slow cooker and add the beans, tomatoes, broth, rosemary, pepper and salt.
Cover and cook on low until the carrots are tender and the broth tastes fully seasoned, 5 1/2 to 6 hours.
Stir in the kale, cover and cook on high until the leaves are tender but still green, 15 to 20 minutes.
Finish with the lemon juice and adjust the salt after tasting the sausage-seasoned broth.