In a large saucepan, combine the broth, water, barley & thyme. Season with salt & pepper; bring to a boil. Cover & cook over low heat until the barley is nearly tender, about 18 minutes.
In a large skillet, heat oil. Add the mushrooms & shallot, season with salt & pepper; cook over high heat until tender and browned, about 8 minutes.
In a medium bowl, combine the ground beef, egg, bread crumbs, cheese, 1/2 teaspoon of salt & 1/4 teaspoon of pepper. Knead the mixture until blended, then roll it into sixteen 1-inch balls.
Add the meatballs & mushrooms to the soup. Simmer over moderate heat until the meatballs are cooked through & the barley is tender, about 8 minutes. Discard the thyme. Stir the parsley into the soup & serve in bowls with sour cream.