Heat oil in a 12 to 14-inch saute pan, over medium heat. Add the onions, garlic and 8 ounces of sliced mushrooms and cook for 5 to 6 minutes. Add the salt and pasta with 1/2 cup pasta cooking water to the pan and toss quickly. Add the remaining 4 raw sliced mushrooms and toss again, off heat. Serve immediately, topped with the grated cheese.
Place both types of flour in a large mixing bowl and stir to mix well. Make a well in the center of the flour mixture and add the water a little at a time, stirring with your hands until a dough is formed. You may need more or less water, depending on the humidity in your kitchen.
Place the dough on a floured work surface and knead it like bread until smooth and elastic, about 8 to 10 minutes. Cover the dough and let it stand for 10 minutes at room temperature.
Roll the dough into long dowels about 1/4 to 1/2-inch thick. Place the pasta strands between 2 hands and lightly roll back and forth to create a lightly spiraled, snake-like noodle. Place the pici on a sheet tray that has been dusted with semolina flour, cover the pasta with a clean dish towel, and set aside until ready to use. At this point, the pasta can be frozen for several months.