Soak 1/4 cup thinly sliced red onion in cold water; set aside. Whisk 2 tablespoons white wine vinegar with 1 small minced garlic clove, 1 teaspoon dijon mustard and a pinch of sugar; gradually whisk in 1/4 cup olive oil. Toss with 1 pound thinly sliced white mushrooms and 1/2 cup chopped parsley. Drain the onion; add to the mushrooms along with 1/2 cup shaved pecorino. Season with salt and pepper and toss. Drizzle with olive oil and top with more cheese.