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Fill a large pot with water and season well with a few big pinches of salt. Bring to a boil.
Meanwhile, do some veg prep! Tear off and discard stems of 1½ lb. crimini or button mushrooms, then tear them into quarters (or in halves if small). Transfer to a medium bowl. Lightly smash and peel 6 garlic cloves, then thinly slice. Peel and finely chop 2 medium shallots. (A small red onion will work fine in a pinch.) Coarsely chop 1 cup parsley leaves with tender stems.
Whisk 5 large egg yolks, 1 large whole egg, 4 oz. store-bought pre-grated Parmesan (about 1¼ cups), and 1½ tsp. freshly ground black pepper in another medium bowl; set aside.
Heat a large Dutch oven over medium-high for a good 3 minutes. You want to get the pan very hot since adding the mushrooms is going to lower the temperature of the surface of the pan. Toss mushrooms and ¼ cup extra-virgin olive oil once in pan to coat in oil. Cook, tossing once every 4–5 minutes (but mostly undisturbed), until mushrooms are mostly golden brown, 13–16 minutes. This will take some time and they will let out a lot of moisture before they start to brown, so stick with it!
Once mushrooms have been cooking for about 10 minutes, drop 1 lb. orecchiette into boiling salted water and set a timer 2 minutes shy of al dente according to package directions.
Back to those mushrooms! Once you’ve finished the browning process, reduce heat to medium-low and add garlic, shallots, and 1½ tsp. salt. Cook, stirring often, until aromatics are softened but not browned, 30–60 seconds.
When pasta is 2 minutes shy of al dente, scoop out 2 cups pasta cooking liquid, then drain pasta.
Add pasta along with 1 cup pasta cooking liquid to mushroom mixture. Cook over medium-low heat, stirring often to finish cooking the pasta and absorb liquid, about 2 minutes (this is why you’re cooking the pasta 2 minutes shy of al dente; it allows for the flavors to meld as the pasta finishes cooking in the sauce). Remove from heat and let cool 1 minute. (Don't skip this step—if the pasta is too hot when you add the egg mixture, it will turn into scrambled eggs instead of a luxurious sauce.)
Add ½ cup pasta cooking liquid to reserved egg mixture and whisk to combine and loosen eggs. Gradually add egg mixture to pot, stirring vigorously with a wooden spoon and adding more pasta cooking liquid as needed to loosen things up, until a very creamy, luscious sauce coats all noodles.
Add parsley and stir again to combine. Taste and adjust seasoning for salt.
Divide pasta among bowls. Top with Parmesan and a few cranks of pepper.