In a bowl, toss the onion with 1 tablespoon of the flour and 1/4 teaspoon salt. Heat 1 1/2 tablespoons of the oil in a large skillet over medium-high heat. Cook the onion, stirring, until golden brown, 1 to 2 minutes; transfer to a plate.
Wipe out the skillet and heat the remaining tablespoon of oil over medium-high heat. Cook mushrooms, stirring, until tender, 5 to 6 minutes.
Reduce heat to medium. Sprinkle the remaining tablespoon of flour over the mushrooms and cook, stirring, for 1 minute. Add the milk and 1/4 teaspoon each salt and pepper. Simmer, stirring, until mixture begins to thicken, 1 to 2 minutes. Add green beans and cook until heated through, 3 to 4 minutes. Top with the onions.