Bring a large pot of heavily salted water to a boil.

Melt the butter and olive oil in a large skillet over medium-high heat until the oil shimmers. Add the shallots and garlic and cook until softened and fragrant, about 2 minutes.

Stir in the shiitake and cremini mushrooms with a pinch of salt. Cook, stirring occasionally, until the mushrooms release their liquid, the moisture evaporates, and the mushrooms become deeply browned, 10 to 12 minutes.

Drop the farfalle into the boiling water as soon as the mushrooms begin browning in the skillet. Cook until the pasta is al dente.

Stir the thyme into the skillet and cook until fragrant. Pour in the vegetable broth, heavy cream, and lemon juice. Simmer the mixture until the sauce reduces slightly, looks glossy, and coats the back of a spoon, about 5 minutes.

Drain the pasta and return it to the pot. Add the mushroom sauce, Parmesan, and parsley, tossing well to coat the noodles evenly. Season with salt and pepper to taste.

Serve immediately while hot, passing extra Parmesan at the table.