Place onion, eggs, salt, and pepper in a large bowl, and beat until the eggs are frothy.
Lay out 4 squares of paper towel on your counter. One at a time, run the matzos under cold water for about 15 seconds on each side (if you break a square, discard it, and start again), and place them side by side on the paper toweling.
Carefully place a matzo square in a shallow dish with sides at least 1 inch high. Spread 1/4 of the onion-egg mixtures on top of it. Pile the other matzos on top, spreading each square with 1/4 of the onion-egg mixtures. Let sit for 5 minutes.
Heat a thin layer of oil in a large skillet. Fry the matzos, one at a time, until well browned on each side. Drain on paper towels.
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