To Prepare the Pasta Dough:
On a large, clean work surface, pour flour in a mound. Make a well in the center about 4 inches wide to cradle the egg mixture. In a medium-sized bowl combine whole eggs, egg yolks, squid ink, and salt. Beat mixture thoroughly with a fork. Pour into flour well. Using your hands gradually incorporate flour into the eggs working from the outside in, until a wet, sticky dough has formed.
Coat your hands in flour and begin folding the dough into a ball by turning the dough a half turn each time. As necessary, add more flour to your hands to keep them from sticking to the dough. Work the dough until it feels elastic, firm and dry. This process should take about 3-5 minutes. Once a smooth ball is formed, wrap ball of dough tightly in plastic wrap and rest on countertop for 30 minutes.
To Prepare the Wild Mushroom Sauce:
While the dough is resting, prepare the sauce. Chop the garlic and the chilies. Use more or less of both as you like.
Heat the olive oil in a pan, add the chopped ingredients and cook for a couple of minutes to soften. The garlic should not get to much colour.
Add the wild mushrooms to the pan, sprinkle with a little salt and pepper and cook, for 5-6 minutes.
To Roll the Pasta Dough:
Once rested, place a sheet of parchment paper on a tray or cutting board and dust lightly with flour. Unwrap rested dough and cut into quarters. Set one quarter on work surface and re-wrap remaining dough in plastic wrap. With a rolling pin, flatten the quarter of dough into an oblong shape about 1/2 inch thick.
Set pasta maker to widest setting and pass dough through the machine at this setting three times.
Returning the dough to your lightly floured work surface, fold both ends in so that they meet at the center. Then fold the dough in half where the end points meet, trying not to incorporate too much air into the folds. Using a rolling pin, flatten dough to 1/2-inch thick. Pass through the pasta maker 3 additional times.
Narrow the setting by 1 notch and repeat twice. The dough should now have passed through the third widest setting. Continue passing the dough through the pasta maker, reducing the thickness by 1 setting each time until it reaches the desired thickness and your dough is very delicate and elastic to the touch, and slightly translucent.
Place rolled dough onto a work surface or baking sheet lightly dusted with flour or lined with parchment paper, folding the dough over as necessary so that it fits; use sprinkled flour or parchment paper between the folds to prevent sticking.
Cover dough with plastic wrap or a kitchen towel to prevent drying. Repeat steps with the remaining dough quarters. If making noodles, cut dough into 12 to 14 inch segments.
To Cut Noodles:
Adjust pasta machine to noodle setting of your choice. Working one dough segment at a time, feed dough through the pasta-cutter. Alternatively, cut folded dough by hand with a chef's knife to desired noodle width.
Divide the cut noodles into individual portions, dust lightly with flour, and curl into a nest. Place on parchment-lined rimmed baking sheet and gently cover with kitchen towel until ready to cook.
Bring a large pot of salted water to a rolling boil. Add pasta, stir gently with a wooden spoon, chopsticks, or a cooking fork, and cook, tasting at regular intervals until noodles are just set with a definite bite, about 1 ½ to 2 minutes. Drain, reserving about ¼ cup of pasta water.
Add cooked pasta to the frying pan with the wild mushroom sauce. Pour in the pasta water. Stir well. Serve with the soft boiled egg on the top sprinkled with salt and pepper to taste. Add lemon zest and grated parmesan as garnish if desired.