Set a pot of water to boil. We’ll come back to this in a bit when it’s time to cook the pasta.
Prepare the mushrooms. Soak the dried porcini mushrooms by covering them in boiling water for 30 minutes. Drain them (but save that rich mushroom broth to use in other recipes).
While the dried mushrooms soak, chop the fresh mushrooms into ½-inch dice (1.27 cm) and chop the shallot. Put the shallots and olive oil in a skillet over medium heat. Cook, stirring frequently, until the shallots are softened and fragrant (not browned).
Add the fresh and rehydrated mushrooms to the skillet. Cook, stirring frequently, allowing the liquid to evaporate from the mushrooms. This will take about 10-15 minutes.
While they cook, you can grill the chicken (if it wasn’t previously cooked). My chicken took 7-10 minutes per side in a stovetop grill pan to reach the minimum safe temp of 165°F (74°C). Season the chicken with salt and pepper and cut into bite-sized pieces.
Set both the mushrooms and chicken aside while you work on the pasta and Parmesan sauce.
Cook the pasta to be al dente according to the directions on the pasta package. While it cooks, make the cheese sauce by whisking the eggs, cream, and Parmesan together. After the pasta has cooked, drain it quickly and stir in the butter so that the noodles don't stick together.
Working quickly, toss the cheese sauce together with the pasta while the pasta is still hot. The cheese should completely melt from the heat of the pasta so that there is no graininess. Don't forget to add some salt and lots of black pepper (if you want).
Next, stir the mushrooms and chicken into the pasta and serve. That's all there is to it!