Elevate cacio e pepe pasta with crispy shiitake mushrooms. The trick to crisping the mushrooms is to start them in a cold dry pan. Heating them this way allows the moisture to release, crisping them up, before adding butter for flavor.
Bring a large pot of salted water to a boil. Cook the pasta according to the package directions until al dente. Reserve 2 cups of the cooking water then drain the pasta.
While the pasta is cooking, arrange the mushrooms in a large skillet in an even layer. Turn the heat to medium-high and let the mushrooms cook, untouched, until they begin to release their liquid, 3 to 4 minutes. Add the thyme, 3 tablespoons of the butter and 1/2 teaspoon salt. Cook and melt the butter, stirring the mushrooms occasionally, until they are a deep golden brown, have released all their liquid and crisp up, 4 to 5 minutes.
Remove the mushrooms to a plate and reduce the heat to medium-low. Add the olive oil and remaining 4 tablespoons butter to the skillet and heat until the butter is melted and the foam subsides, about 1 minute. Add the 1 tablespoon plus 1 teaspoon pepper and cook, stirring, until lightly toasted, about 1 minute. Add the pasta and 1 1/2 cups of the reserved cooking water to the skillet and toss to coat. Sprinkle the 2 cups Parmesan on top and continue to cook, tossing until the cheese is melted and the pasta is well coated, about 2 minutes. If necessary, add a little more of the reserved cooking water to loosen the sauce. Transfer the pasta to a large bowl and top with the mushrooms. Garnish with more pepper and Parmesan, if desired.