Heat 1 tablespoon oil in a large skillet over medium-high heat. Cook half the mushrooms with 1/2 teaspoon salt, stirring occasionally, until deep golden, about 5 minutes, and transfer to a bowl. Repeat with 1 tablespoon oil, remaining mushrooms, and 1/2 teaspoon salt. Cook shallot in 1 tablespoon oil over low heat until softened. Add 2 tablespoons vinegar and simmer until reduced by half.