Heat the olive oil in a 6-quart Dutch oven over medium-high heat. Add the chicken sausage and cook for 6–8 minutes, breaking it into small crumbles, until lightly browned and cooked to 165°F.
Add the onion, carrots, and celery. Reduce the heat to medium and cook for 5–7 minutes. Stir in the garlic, tomato paste, Italian seasoning, red pepper flakes, salt, and black pepper and cook for 1 minute.
Pour in the diced tomatoes and 5 1/2 cups chicken broth, scraping any browned bits from the bottom of the pot.
Mash 1 cup of the cannellini beans with the remaining 1/2 cup broth. Stir the mashed beans and remaining whole beans into the soup.
Bring to a gentle boil, reduce to a steady simmer, and cook uncovered for 12–15 minutes, until the carrots are tender and the broth has slightly thickened.
Stir in the kale and simmer for 5–7 minutes. Turn off the heat and stir in the lemon juice and Parmesan. Taste, adjust the seasoning if needed, and serve hot.