Melt half the butter in a large skillet. Add the onions and cook slowly until caramelized, about 20 minutes. Remove the onions. Sauté the mushrooms in the oil and season with salt and pepper; add them to the onions.
Soak the matzos in warm water for 30 seconds, then crumble them into the egg mixture. Add the onions and mushrooms. Season with salt and pepper.
Heat the remaining butter in the skillet. Add the matzo mixture and stir over moderately high heat until the eggs set, about 5 minutes. Sprinkle with the parsley and serve.