Place 2 pots of water on to boil: One medium saucepan with 1 1/2 cups of water in it, one pasta pot with a few inches of water in it. Cover both pots and bring all the water to a boil. When the saucepan with 1 1/2 cups of water comes to a boil, stir in rice and return water to a boil. reduce heat to simmer. Place colander over the pot and add the edamame to it. Place the pot cover over the edamame, nesting it in the colander. Steam the edemame 5 minutes, then remove them to 2 small bowls, salt them and cover bowls with plastic wrap or alum foil to keep warm. Set colander aside and return pot lid to rice, stirring rice before replacing the lid.
While the edamame are steaming, make the gyoza: Add 1/2 cup cabbage to the second pot of water. Blanch the shredded cabbage 1 minute and remove with tongs to paper towels to drain and cool. Chop shredded blanched cabbage. Combine pork, chopped shrimp, sake, ginger, scallion, tamari, a few grinds of black pepper, and the cooled chopped cabbage in a bowl. Place 2 t of filling onto each won ton wrapper. Wet your fingertips to help seal them. The gyoza should look like small half moons.
Place a whole cabbage leaf into the bottom of the colander. The leaf will prevent the dumplings from sticking to the colander. Arrange dumplings on cabbage leaf in colander and steam over second, larger pot of simmering water. Cover and steam, 10-12 minutes. While you are working on this, go back and forth with the edamame as necessary per above instructions.
While the dumplings cook, make the steaks. Slice them thinly across the grain. Toss with seasoning and teriyaki. Heat a nonstick skillet over high heat. Add oil and the meat, and stir-fry. When meat browns at edges, add scallions and cook 2 minutes more, stirring frequently.
Use remaining raw shredded cabbage as a bed to serve your dumplings on. Plate all of your items on little dishes and in small bowls and fit them into your boxes.
Mix the dipping sauce ingredients for the gyoza and set out alongside dumplings.