In a fine-mesh strainer set over a large bowl, combine cabbage and salt. Let sit for 10 minutes.

Transfer cabbage to a clean kitchen towel. Holding towel over the bowl, wring out as much water as possible. Pour liquid out of the bowl and wipe dry.

In the same bowl, combine ground beef, soy sauce, ginger, sesame oil, cabbage, and 1/4 cup of sliced scallions (save the remaining scallions for serving). Using clean hands, mix until well combined and the mixture sticks together.

In a small bowl, beat egg with a fork until blended. Arrange wonton wrappers on a baking sheet in a single layer. Brush the center of each wrapper with a thin layer of beaten egg. Add 1 heaping tablespoon of the beef filling to each wrapper, pressing it into an even circle with a fork that almost reaches the edges of the wrapper.

In a large heavy skillet over medium heat, heat 2 tsp. of oil, swirling the skillet to coat. Working 4 at a time, arrange dumplings in the skillet, filling side down. Cook until browned on the bottom, about 30 seconds. Flip to wrapper side, add 1 tsp. water, and cover the pan. Steam until the filling is cooked through and the wrapper is crisp and browned on the bottom, about 30 seconds more. Transfer to a platter. Repeat with the remaining filled wrappers and oil.

Top the smash dumplings with reserved scallions. Serve with soy sauce on the side for dipping.