Preheat the oven to 225°. In a large skillet, melt 1 tsp. of the butter.

Add the shiitake and cook over medium high heat, stirring until golden, about 5 minutes. Add the shallots and cook, stirring, until softened, about 2 minutes. Keep warm.

Crack 2 eggs into a medium bowl, season with salt and pepper and beat with a whisk.

In a 10-inch, safe nonstick skillet, melt 1/2 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-fourth 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 remaining omelets.