Make the dumplings first, as they need to dry. Mix the egg, salt and water together. Put the flour into a bowl and pour the egg mixture in. Mix well until a dough forms, then knead it for 5 minutes. Shape the dough into a ball and then flatten it out somewhat to make a fat disk. Let this dry while you start the soup.
Chop the duck carcass into chunks using kitchen shears or a large knife or cleaver. Put it into a large pot with the water and turn the heat to medium. It is important to never let this soup boil, or it will turn very cloudy and can get bitter.
Heat the duck fat in a sauté pan and cook the onions over medium heat, stirring occasionally. You want to caramelize the onions here, so as you cook them you may need to drop the heat to medium-low and cover if need be. Take your time doing this, as the flavor of the caramelized onions is important. You want the onions to be well browned.
Make the dumplings by running the dough against the large edge of a box grater. Hold the dough in one hand, the grater in the other over a tea towel or other clean cloth, and be sure to move the grater so all the dumplings don’t land on each other and make a doughy mess. You will need to flip the dough a lot so you have a good edge that will grate properly. They should look like raggedy spaetzle. Let the dumplings dry on the cloth as the soup cooks.
Add the bay leaf, black pepper, juniper and dried mushrooms to the soup. Let this cook gently while the onions finish caramelizing. When the onions are ready, move them to the soup and add the remaining soup ingredients. Let this simmer very gently, uncovered, for 90 minutes.
Turn the heat off the soup and strain it. Start by fishing out all the big pieces of duck and vegetable with a slotted spoon or a Chinese spider skimmer. Then strain the soup through a fine-mesh strainer into another pot. If you want to get fancy, strain this again through the fine-mesh strainer, only this time with cheesecloth or a plain paper towel set inside the strainer; this filters out much of the fat, and very fine debris. You may have to change the paper towel midway through: It blots up a lot of fat, and this can stop the soup from straining. Add salt to taste and set the strained soup in a pot over low heat to keep warm.
Bring a large pot of salted water to a boil. Drop the dumplings in a few at a time and boil until they float on the top, then for another minute. Skim them out and lay the cooked dumplings on a baking sheet while you finish cooking the rest. To serve, put some dumplings in each person’s bowl and ladle over the soup.