Heat the butter and oil in a 12-inch skillet over medium heat until the butter has melted and the foam subsides. 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 begin softening. Stir in the cremini mushrooms and salt; cook, stirring frequently, until the mushrooms have released their moisture and turned golden brown, about 8 minutes.

Stir in the thyme and cook until fragrant. Transfer the mushroom mixture to a bowl. Add the broth to the empty skillet and bring to a boil, scraping the bottom of the pan with a wooden spoon to release the browned bits. Remove the skillet from the heat, stir in the cream and lemon juice, and season with salt and pepper to taste.

Boil the pasta in 4 quarts of salted water until al dente. Reserve 1 cup of the pasta cooking water, then drain the pasta and return it to the pot. Stir in the mushroom mixture, the cream sauce, Parmesan, and parsley.

Cook the mixture over medium-low heat, tossing constantly to combine, until the pasta absorbs most of the liquid and the sauce clings to the noodles, about 2 minutes. Adjust the consistency with the reserved cooking water if the sauce becomes too thick. Serve immediately with extra Parmesan.