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