Heat the vegetable oil in a large pot or Dutch oven over medium-high heat. Add the sausages and cook, stirring occasionally, until lightly browned and crisp, about 5 minutes. Transfer to a plate with a slotted spoon.
Add the celery, carrot and onion to the pot and cook, stirring occasionally, until slightly softened, about 5 minutes. Season with salt and pepper. Add the cherry tomatoes, white beans (and their liquid), baked beans, chicken broth and 2 1/2 cups water. Bring to a boil, then reduce the heat to medium and continue cooking until the soup thickens slightly and the tomatoes just start bursting, about 20 minutes.
Return the sausages to the pot along with the parsley; season with salt and pepper. Top each serving of soup with the cheese.