Skillet Gnocchi with Sage Brown Butter Mushrooms

Name: Skillet Gnocchi with Sage Brown Butter Mushrooms
Ingredients: 1 pound fresh mushrooms, such as cremini, shiitake, or oyster mushrooms, or a combination; 1 small bunch fresh sage; 2 tablespoons olive oil, divided; 2 (16- to 18-ounce packages) fresh, shelf-stable, or frozen potato gnocchi (do not thaw), divided; 7 tablespoons unsalted butter, divided; 1 teaspoon kosher salt, divided; 1/4 teaspoon freshly ground black pepper, divided; 3 tablespoons balsamic vinegar; Freshly shaved Parmesan or Pecorino Romano cheese, for serving