Bring a large pot of water to a boil.
Whisk 4 egg yolks and 2 eggs in a medium bowl until no streaks remain, whisk in ½ cup Parmesan; season with pepper and set aside.
Heat 2 tablespoons oil in a large Dutch oven or pot over medium-high. Add 8 ounces mushrooms and cook, stirring occasionally, until golden brown, about 10 minutes. Transfer mushrooms to a plate and set aside.
Season boiling water with 2 tablespoons kosher salt, add 1 pound pasta and cook 2 minutes less than package instructions, stirring occasionally.
Measure out 1 ¾ cups pasta cooking water. Add 1 cup cooking water and 2 tablespoons truffle butter to Dutch oven; bring mixture to a boil over medium-high.
Drain pasta then transfer pasta to Dutch oven.
Cook pasta, stirring and gently tossing constantly with tongs, until al dente and water is reduced by about half; about 2 minutes. Remove Dutch oven from heat.
Whisk ¼ cup reserved pasta cooking water into reserved egg mixture. Slowly stream egg mixture into pasta, stirring and gently tossing constantly with tongs, until cheese is melted and egg is thickened to form a glossy sauce on the pasta.
Thin sauce with remaining ½ cup reserved pasta cooking water, adding a 1–2 tablespoons at a time, until consistency of heavy cream is reached. Season to taste with additional truffle butter and salt.
Gently toss in reserved mushrooms and divide pasta among serving bowls. Top with thin shavings of truffle (if using), ground pepper and additional Parmesan.