In a medium skillet, heat the oil until shimmering. In a small bowl, beat the eggs. Spread the panko in a shallow bowl. Dip each mushroom into the egg and let the excess drip off, then coat thoroughly with panko. Fry the mushrooms, 4 at a time, over moderately high heat until golden brown, about 2 minutes per side. Drain on a wire rack set over a baking sheet. Season with salt and serve right away.