Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Stir in orecchiette and return to a boil. Cook uncovered, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Reserve 2/3 cup pasta water prior to draining pasta.
Heat 2 tablespoons butter, olive oil, and a pinch of salt in a deep frying pan over medium heat. Add onion; cook and stir until soft but not brown, 3 to 4 minutes. Stir in remaining 2 tablespoons butter and mushrooms; cook until tender, 5 to 7 minutes more. Add garlic and sage and cook until soft, about 2 minutes. Pour in vermouth; bring to a boil, then reduce to heat to low. Stir in Parmesan cheese until melted, 2 to 3 minutes.
Mix in cooked pasta and reserved pasta water. Toss pasta to coat well. Season with salt and pepper. Top with parsley and serve.
