Peel and thinly slice the carrots. In a large saucepan, begin heating the chicken broth and carrots. Bring to a boil and then drop the temperature down to a simmer.
Meanwhile, cut the onion into halves or quarters and slice so that you have long slices of thin cut onion. When are just barely getting tender, drop in the onions and stir. Continue simmering.
Peel, core, and chop the apples. When onions are softened, drop in the apples. Heat until carrots and onions are completely soft and apples are warmed through and slightly softened.
Place one slice of cheese into each of four individual bowls. Ladle soup equally into each bowl. As the heat from the soup melts the cheese, stir the soup to mix it around. If cheese does not melt evenly, you might try ripping the cheese into smaller pieces before mixing inches.