Great question — answering as the UphillTech team:
The eggs are added at different times to control the dough’s texture. The first egg, added while the dough is warm, binds it and keeps it smooth. The second egg, added after cooling, prevents the dough from getting too thin. This method helps create light, fluffy dumplings without making the mixture too soft or runny.
Full answer by Mike Green: https://cooking.stackexchange.com/a/130418 (CC BY-SA 4.0).
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