Heat the oil in a medium skillet over high heat until hot. Reduce the heat to medium; add the onions and cook, stirring occasionally, until golden, about 8 minutes. Season with salt and pepper, to taste, and transfer to a bowl with a slotted spoon. Reserve the skillet and oil in which the onions were cooked.
Meanwhile squeeze the spinach well, a handful at a time to remove all the water and set aside. Break up the matzos slightly and soak them in a bowl of cold water for 3 minutes. Drain the matzos and combine them with the eggs, 3/4 teaspoon salt, and 1/4 teaspoon pepper.
Reheat the oil remaining in the reserved skillet with the butter over medium-high heat until hot. Add the matzo mixture and cook, stirring constantly, until eggs are scrambled and the matzo has begun to crisp, about 5 minutes. Stir in the spinach and cheese and cook until just heated through. Top each portion with a mound of the onions.