How to Make It
Melt 1 tablespoon butter in a large skillet (not nonstick) over medium. Add sliced onion and 1/2 teaspoon salt; cook, stirring occasionally, until translucent, about 8 minutes. (If you insist, you can use oil instead of butter to cook the onions.) Reduce heat to low, and cook, stirring occasionally, until onions are fragrant and brown, about 45 minutes. (Watch carefully to keep onions from burning.) If the onions start to stick, add 1 tablespoon of water from time to time and allow it to evaporate.
Break matzos into a strainer or colander until pieces are fairly small (about 1 inch); discard any crumbs that fall through strainer. (If you’re cooking with a child, let them do this; smashing things up is very satisfying work.) Briefly run matzo pieces under running tap water until just moistened; transfer to a medium bowl. Add eggs and caramelized onions, and mix thoroughly.
Wipe skillet clean, and melt remaining 1/4 cup butter over medium-high. When butter is sizzling, add matzo mixture and sprinkle with remaining 3/4 teaspoon salt. Cook, stirring occasionally, until eggs are set and you have some golden crisp bits, 3 to 4 minutes. Serve to 4 very happy people.