Break the eggs into a large bowl. Add ¾ teaspoon salt and a generous amount of pepper and whisk to blend. Mix in 1 cup of the Parmesan cheese.

Heat 3 tablespoons of the oil in a heavy 12-inch heavy skillet (preferably nonstick) over medium-high heat. Add the onion and sauté until beginning to brown, about 5 minutes. Add the sage and then the shiitake. Sprinkle with salt and pepper and sauté until the mushrooms are tender, about 4 minutes. Add the spinach and stir until beginning to wilt, about 1 minute.

Add the remaining 1tablespoon oil to the skillet and then the egg mixture. Stir gently to distribute evenly. Reduce the heat to medium-low. Sprinkle the goat cheese over the eggs. Cover and cook until the eggs are almost set but still moist in the center, about 10 minutes.

Meanwhile, preheat the broiler.

Sprinkle the remaining ½ cup Parmesan cheese over the frittata. Broil until the eggs puff, the center is just springy to the touch, and the cheese begins to brown, about 3 minutes. Using a flexible rubber or silicone spatula, free the frittata around the edges. Slide onto a platter or leave in the pan. Sprinkle with the parsley or thyme and sere hot, warm, or at room temperature, cutting into wedges.