Although it makes a delightful breakfast, the frittata could also be served with a green salad for a lunch or light dinner.
Preheat oven to 350 degrees. Heat olive oil in a medium ovenproof skillet over medium heat. Add scallions, mushrooms, and 1/2 teaspoon salt; cover, and cook until the vegetables are very soft, about 15 minutes. (The vegetables start out piled high, but they cook down quickly.)
In a large bowl, whisk together eggs and 1/4 teaspoon each salt and pepper. Pour egg mixture over the vegetables, stir for 2 minutes, then cook for 3 to 4 minutes more. Sprinkle cheese over the frittata, and bake in the oven for 15 minutes.
Cleaning shiitakes: Remove the mushroom stems and discard, then gently wipe the caps with a damp paper towel.
White button mushrooms can be substituted for the shiitakes; trim but do not remove the stems.