For the pasta dough:
Combine the 00 and semolina flours in a mixing bowl and then pour them into a pile on a clean flat work surface. Make a well in the center of the flour.
Pour the eggs into the well, and then beat with a fork until the eggs are smooth. Slowly start dragging the edges of the flour into the center of the well with the fork to combine the flour with the eggs until the dough starts to come together. Form the mixture into a ball with your hands and knead until smooth, about 10-15 minutes. Cover the dough with plastic wrap or a towel and let it rest for at least an hour.
For the cappelletti:
To make the cappelletti filling, place the ricotta cheese, goat cheese, 1 teaspoon salt, pepper and red chili flakes in a food processor fitted with a metal blade (or blender) and pulse just until combined. Place mixture in a piping bag or a small mixing bowl.
Generously dust a large sheet pan with flour or cornmeal. Remove the dough from the plastic wrap and cut into four pieces of roughly equal size. Work with one piece at a time and keep the rest covered with plastic wrap.
Starting on the widest setting, run the dough through a pasta machine. Continue stretching pasta dough, one setting at a time, until dough is about 1mm thick (typically around setting #6 on a pasta machine). Lay the pasta sheet flat on a lightly floured surface. Alternately, roll dough using a rolling pin until is about the same thickness.
Use a 3-inch diameter circle biscuit cutter or glass to stamp circles in the sheet of dough. Remove the excess dough surrounding the circular stamps. Ball up the excess dough, cover it with plastic wrap or a towel and set aside.
Pipe or spoon about 1 teaspoon of filling into the center of each pasta dough circle. Fold the bottom of the dough circle to the top to create a semi-circle. Press the seam tightly while pressing out all the air. Lightly wet the top of the circle with your fingers if the dough is not sticky enough to create a seam. Rotate the semi-circle 180 degrees so that the side of the semi-circle without the seam is at the top. Use your finger to press a small dent into the dough. Grab the two corners of the bottom of the folded semi-circle and press them together tightly to form a cappelletti. Transfer the shaped cappelletti to the sheet pan. Repeat the forming process with all remaining dough.
For the sauce:
Fill a stockpot 2/3 full of water. Salt the water heavily and bring the water barely to a boil.
Meanwhile, heat olive oil in a large Dutch Oven or pot set over medium heat. Add the garlic and chiles and cook until fragrant, about 2 minutes. Add the mushrooms and stir frequently until most of the water in the mushrooms has evaporated. Add the butter, miso, soy sauce and pepper and stir to combine. Add the heavy cream and whisk until sauce thickens slightly. Reduce heat to low.
Add the cappelletti to the boiling water in the stockpot. Cook the cappelletti for about 5 minutes until al dente, or desired texture. Reserve about 1/2 cup of the pasta water and then drain the cappelletti using a colander.
Add 1/4 cup of the pasta water to the sauce. Whisk the sauce until it is glossy. Add the cappelletti to the sauce and stir until the pasta is coated, adding additional pasta water or cream if needed to create a smooth, glossy consistency. Season to taste with additional salt and pepper if desired, and serve immediately garnished with parmesan cheese.
