Bring a large saucepan of salt water to boil (about 1 teaspoon of salt) over medium-high heat. Once the water has come to a boil add the pasta and cook according to the package directions until it is al dente. Drain pasta reserving one cup of pasta water.
Add olive oil and 1 tablespoon of the butter to a large skillet over medium-low.
Once the butter has melted add shallots and a pinch of salt. Sauté until they are soft and golden. Stir occasionally to prevent them from burning.
To the cooked shallots add the trumpet and cremini mushrooms along with a pinch of salt and cook until they have released their water and have cooked down but are still firm about 5-7 minutes. Stir occasionally to prevent them from burning.
Add the thyme to the mushrooms and shallots along with the shiitake mushrooms and cook for another 2-3 minutes.
Add the dry sherry and broth. Allow the sauce to reduce by half.
Add the remaining ½ tablespoon of butter and heavy cream, reduce to low, and continue to cook for 3 minutes.
Add the flour along with ¼ cup pasta water and combine well to form the sauce. Add additional pasta water as needed to get a creamy consistency.
Serve mushroom sauce over pasta and top with grated Pecorino Romano and parsley.