Heat the olive oil in a 4-quart braiser or large, wide pot over medium heat. Add the chicken sausage links and cook, turning occasionally, until nicely browned on all sides. They do not need to be completely cooked through. Transfer to a cutting board. Once cool enough to handle, slice into approximately ½-inch rounds.
Add the sliced onion, carrots and celery to the same pan. Cook over medium heat for about 8 minutes, stirring occasionally and scraping up any browned bits left from the sausage.
Add the garlic, tomato paste, dried thyme and red pepper flakes. Cook for about 1 minute, stirring frequently, until the garlic is fragrant and the tomato paste begins to darken.
Add the whole and halved grape tomatoes. Cook for 2–3 minutes, stirring occasionally, until the halved tomatoes begin to soften and release some of their juices.
Add the butter beans and chicken bone broth. Use the back of a wooden spoon to mash a generous spoonful of the butter beans against the side of the pan, then stir the mashed beans into the broth. Add the rosemary sprig and Parmesan rind.
Bring to a gentle simmer. Reduce the heat as needed and simmer uncovered for about 20 minutes, stirring occasionally.
Add the sliced sausage. Continue to simmer uncovered for 15–20 minutes, or until the sausage is cooked through and the carrots and celery are tender.
Remove and discard the rosemary sprig and Parmesan rind. Stir in the baby kale and cook for 3–5 minutes, until wilted and tender.
Season with salt and freshly ground black pepper to taste.
Spoon into bowls and top with freshly grated Parmesan. Serve with sourdough bread, if desired.