Heat the butter and oil in a 12-inch skillet over medium heat until the butter has melted. Add the shallots and cook until softened and translucent, about 4 minutes.

Stir in the garlic and cook until fragrant. Add the shiitakes, increase the heat to medium-high, and cook for 2 minutes to give them a head start.

Add the cremini mushrooms and salt. Cook, stirring often, until the mushrooms have released their moisture and turned golden brown, about 8 minutes.

Stir in the thyme and cook until fragrant, about 30 seconds, then transfer the entire mushroom mixture to a bowl.

Pour the broth into the empty skillet and bring to a boil, using a wooden spoon to scrape up the browned bits from the bottom of the pan. Remove from the heat, stir in the cream and lemon juice, and season with salt and pepper.

Bring 4 quarts of water to a boil in a large pot. Add the pasta and salt and cook until al dente. Reserve 1 cup of the cooking water, then drain the pasta and return it to the pot.

Stir the mushroom mixture, broth mixture, Parmesan, and parsley into the pasta. Cook over medium-low heat, tossing constantly, until the pasta is coated and has absorbed most of the liquid, about 2 minutes. Add the reserved cooking water as needed to loosen the sauce to a silky consistency.

Serve immediately with extra Parmesan passed at the table.