Place the mushroom in a bowl and pour boiling water over the top and set aside.
Cook the noodles according to the package instructions and set aside.
Season the duck breast with the salt.
Place skin side down on a cold heavy bottomed skillet (preferably cast iron).
Cook for 5-7 minutes on medium heat, until fat is golden brown.
Flip over and cook 1 minute then remove from heat and let cool.
Heat the duck stock in a Dutch oven and add the soy sauce, vinegar, brown sugar and mirin. (you will need to add additional salt if using unsalted duck stock)
Bring to a simmer for 10 minutes. Adjust salt to taste if needed.
Squeeze the excess liquid from the mushrooms and slice.
Add ½ of the mushroom cooking liquid to the broth and remove from heat.
Slice the duck breast with a sharp knife.
Add the broth in bowls with the cooked noodles, sliced duck breast and shiitake mushrooms.
Garnish with ramen eggs, crispy garlic, and chili oil.