Trim ends off brussels sprouts, remove any damaged external leaves, and cut in half lengthwise. Set aside.
Remove casing from sausage and cut into 1/4-in.-thick slices. Put olive oil and sausage in a medium pot over medium-high heat and cook, stirring occasionally, until sausage is well browned, about 7 minutes.
Meanwhile, peel potatoes and cut into roughly 1/2-in. pieces.
When sausage is browned, add bay leaves and caraway seeds to the pot. Cook, stirring, until fragrant, about 1 minute. Add potatoes, broth, and 1 cup water. Bring to a boil. Add brussels sprouts. Partially cover pot and reduce heat to low or medium-low to maintain a steady simmer. Cook soup until potatoes and brussels sprouts are tender, about 15 minutes. Remove bay leaves and serve soup hot.
Note: Nutritional analysis is per serving.