In a bowl, combine meat, cabbage, ginger, garlic, scallion whites and 2 tablespoons soy sauce and 1/4 cup water.
Lay a wrapper on a clean, dry surface, and, using your finger or a brush, spread a bit of egg along half of its circumference.
Place a rounded teaspoon of filling in the center, fold over and seal by pinching edges together; do not overfill.
Place dumplings on a plate; if you want to wait a few hours before cooking, cover plate with plastic wrap and refrigerate; or freeze, for up to two weeks.
To cook, put about 2 tablespoons oil in a large non-stick skillet and turn heat to medium-high.
A minute later, add dumplings, one at a time; they can touch one another, but should still sit flat in one layer.
Cook about 2 minutes, or until bottoms are lightly browned and most of the oil has been absorbed.
Add 1/4 cup water, per dozen dumplings, and cover.
Lower heat to medium and let simmer about 3 minutes.
To make the dipping sauce, combine remaining soy sauce, green parts of scallions and vinegar.
Uncover dumplings, return heat to medium-high and cook another minute or two, until bottoms are dark brown, crisp and water evaporates; use more oil if necessary.
Serve hot, with sauce.
