Bring a large pot of salted water to a rolling boil. Add the pasta and cook until al dente according to the package directions. Reserve 1 cup of the starchy cooking water before draining.

While the pasta cooks, heat the olive oil and butter in a large skillet over medium-high heat until the butter melts.

Add the shallots and garlic. Cook for 2 to 3 minutes, stirring frequently, until softened and fragrant.

Add all three types of mushrooms to the skillet. Toss to coat with the oil, then season with the rosemary, salt, and pepper. Cook for 7 to 10 minutes, stirring occasionally, until the mushrooms are deeply golden and their juices have evaporated.

Stir in the tomato paste and cook for 1 minute until it darkens slightly. Pour in the red wine and 0.5 to 0.75 cup of the reserved pasta water. Simmer over medium heat for 4 to 5 minutes until the sauce reduces and thickens.

Add the drained pasta to the skillet. Toss to coat, adding more pasta water a splash at a time if the sauce looks tight.

Remove from heat. Stir in the Parmesan. Serve topped with the chopped parsley, walnuts, and a pinch of red pepper flakes if desired.