How to Make It
Preheat the oven to 225°. In a large skillet, melt 3 tablespoons of the butter. Add the shiitake and cook over moderately high heat, stirring occasionally, until golden, about 7 minutes. Add the shallots and cook, stirring, until softened, about 3 minutes. Add the pea shoots, season with salt and pepper and cook just until wilted, about 1 minute. Keep warm.
Crack 6 eggs into a medium bowl, season with salt and pepper and beat with a whisk. In a 10-inch, nonstick skillet, melt 1 tablespoon of the butter. Whisk the eggs again and add them to the skillet. Cook over moderately high heat, lifting the edges with a spatula to allow the uncooked eggs to seep underneath, until the bottom of the omelet is golden and the top is nearly set, about 4 minutes.
Spoon one-third of the filling down the center of the omelet and sprinkle with one-third of the goat cheese. Using a rubber spatula, fold the sides over the filling to enclose it completely. Slide the omelet onto a large heatproof plate and cut it in half; serve immediately or transfer it to the oven to keep warm. Repeat with the remaining butter, eggs and filling to make 2 more omelets. Serve at once.