For the gnocchi: Preheat the oven to 375 degrees F. Wash the potatoes and, while still wet, coat with salt. Bake until fork tender, 45 minutes to 1 hour. Cut the potatoes in half, then scoop out the flesh and run through a ricer or food mill.
Place the potatoes on a clean surface in a mound, and then make a well in the center. Place the flour and egg in the center of the well, and gently mix the flour and egg into the potatoes with a fork, being careful to avoid over-mixing.
Bring the dough together gently with your hands, and then roll it out into tubes. Cut into 1-inch pieces to form the gnocchi.
Chill the gnocchi until firm, 1 hour.
Bring a medium pot of water to a boil. Add the gnocchi and cook until they float, 5 minutes, then drain.
For the sauce: In a pan set over medium heat, add the butter, onions and garlic. Saute until tender, 5 minutes. Add the mushrooms and saute until tender, 5 minutes. Add the cream and porcini powder and bring to a boil, then reduce to a simmer for 5 minutes. Remove from the heat and slowly add the egg yolk and mix to temper. Return the pan to the heat and bring to a light simmer to thicken the sauce. Add the gnocchi and bring the sauce back to a simmer.
To serve, plate the gnocchi and garnish with the Parmesan and basil.