Cook the pasta in a large pot of boiling water with 1 tsp salt, until al dente, according to package instructions. Reserve 1 cup of pasta cooking water, drain, and keep warm.
Make the sauce while the pasta is cooking. Melt the butter and olive oil in a large skillet over medium-high heat. Brown the mushrooms for 2-3 minutes on each side. Add the garlic, parsley, and red pepper flakes.
Pour in the wine, deglaze the pan, and simmer until reduced by half.
Mix in the chicken broth, heavy cream, and Parmesan cheese. Stir until the cheese has melted and the sauce has thickened slightly. Check the seasoning and add ½ tsp of salt if needed, and freshly ground black pepper to taste
Add the cooked pasta to the skillet and toss until well coated with the sauce. If you feel the sauce is too thick, add some of the pasta cooking water a tbsp at a time, until it reaches your desired consistency.