Heat a deep, 12-inch sauté pan or very wide pot over medium heat, then add oil. Add shallot and ginger; cook, stirring occasionally, until shallot begins to brown. Add mushrooms and 1 teaspoon water, reduce heat to medium-low, and cover pan to allow mushrooms to sweat, about 5 minutes. Once mushrooms have begun to soften, remove lid, increase heat to medium, and cook until mushrooms brown, about 10 minutes. Scrape mixture into a bowl and set aside.