Increase the heat to medium-high and add the remaining 1 teaspoon oil to the pan. Add shallot (or onion) and cook, stirring, until fragrant, about 15 seconds. Add mushrooms and cook, stirring occasionally, until softened and browned, 1 to 2 minutes. Sprinkle the mushrooms with the remaining 1 tablespoon flour and cook, stirring, for 30 seconds. Pour in broth and wine and bring to a boil, stirring constantly. Reduce heat to a steady simmer and cook, stirring occasionally, until slightly thickened, about 5 minutes.