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In a small bowl, mash the goat cheese with the garlic and chives.
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.
In a 12-inch nonstick skillet, heat 2 teaspoons of the olive oil. Whisk the egg whites until frothy and season with salt. Add half of the beaten egg whites to the skillet and cook over moderately high heat, stirring gently, until golden on the bottom and the top is nearly set, 2 to 3 minutes. Spoon half of the mushroom mixture onto one side of the omelet, and using a spatula, fold over the opposite side. Slide the omelet onto a large plate, cut it into 3 wedges and serve immediately. Repeat with the remaining olive oil, egg whites and mushrooms.
The mushroom mixture can be refrigerated overnight. Reheat gently before using.