Heat a medium-sized saucepan over medium heat until hot. Add the avocado oil and swirl the pan. Add the kielbasa that has been sliced on a slight bias. Brown the sausage in the pan for about 2-3 minutes on medium heat or until nicely brown on both sides.
Next, remove the sausage to a bowl. Add in diced white onion, minced garlic cloves, thinly sliced carrot, and chopped celery stalks. Add a pinch of salt to the vegetables and stir until browned for about 1-2 minutes on medium heat. Add a bay leaf as well. The ingredients above list a crumbled bay leaf but add one whole bay leaf as it is easy to remove.
Add in the chicken broth and 2 cups of water. Bring the mixture to a low simmer and cook for 5-8 minutes. The vegetables should be half cooked at this point.
Add in the kale and white beans. Simmer until the kale is wilted and the beans are heated. Serve immediately. Feel free to top the soup with chopped parsley.