For the gnocchi: Put the porcini in a large bowl and add enough hot water to cover the porcini. Allow the porcini to soak for a few minutes or until soft. Then finely chop the porcini to a paste-like consistency.
On a work table, peel and mash the potatoes with the back of a fork or use a potato ricer. Add the porcini to the mashed potatoes and then add the flour, Parmigiano, salt, and pepper. Combine together and knead until the mixture becomes regular dough consistency, smooth and slightly sticky. Add more flour if needed, which will absorb any excess moisture.
On a lightly-floured work surface, take pieces of the dough and roll them out into thin logs, approximately 1/2-inches thick. Cut them 1-inch long.
Make the ridges on each gnocchi by using the back of a fork in a rolling action.
Cook's Note: The imprints will allow the sauce to get trapped on the surface of the gnocchi.
Put the gnocchi on a tray in a single layer and place a cloth over the top to prevent them from drying out. They may also be set in refrigerator or freezer until needed.
Cook the gnocchi in a pot of boiling salted water for about 1 to 2 minutes or until they float to the surface.
For the sauce: Melt the butter in a saucepan and add the fresh sage leaves. Remove the gnocchi from the boiling water with a slotted spoon or drain and add them directly to the saucepan with about 1/2 cup pasta water. Cook the gnocchi with the sauce for about 30 seconds and then remove them from the heat. Add the Parmigiano and mix well. Plate and serve.