In a large deep skillet, heat 2 tablespoons of the oil until shimmering. Add the mushrooms and cook over high heat, without stirring, for 2 minutes. Add the shallot and cook, stirring occasionally, until the mushrooms are tender and browned, 5 to 6 minutes. Add the white wine and cook, scraping up any browned bits, until evaporated, about 1 minute. Stir in the scallions and cook for 1 minute. Off the heat, stir in the goat cheese and season with salt and pepper.