Remove the stems from the shiitakes. Slice the mushrooms and set aside. In a large pot place the mushroom stems and all of the ingredients up to the whole cloves. Bring to a boil and reduce heat to a low simmer. Skim any foam from the top. Cook the liquid for 4 hours until it has reduced to 2-2 1/2 quarts (about 1/2).
Strain off the liquid, season with salt, pepper, and vinegar. Put the liquid back on the stove. Pull the meat from the bones, season with salt and pepper and set aside. There should be 1 1/2 -2 cup of meat. Bring a small pot of water to a boil. Cook the eggs in the water for 6 minutes. 8-10 for harder cooked eggs. Cool the eggs in water. Peel the eggs.
While the eggs are cooking bring the stock back to a boil. Add the ramen noodles, sliced shiitake, and duck meat. Cook until everything is hot. Turn off the heat and serve. Garnish with eggs, green onion, ginger, and chili flake.