Heat 1 tablespoon of vegetable oil in a large skillet over medium-high heat. Add cabbage and cook, tossing often, until edges are browned, about 4 minutes. Reduce heat to low and continue to cook, tossing often, until thickest parts of cabbage leaves are tender, about 4 minutes longer. Remove from pan and place on a separate plate. Set aside.
Heat remaining tablespoon of vegetable oil in same skillet over medium-high heat. Add pork, breaking up and spreading across surface of pan with a spatula or tongs. Cook pork, undisturbed, until underside is brown, about 3 minutes. When pork is browned, break up meat into small bits. Cook, tossing, just until there’s no more pink, about 1 minute.
While the meat is browning (and you're not touching it), boil udon noodles according to the package. Drain in a colander. Transfer noodles to a bowl and toss with sesame oil.
To your meat mixture, add the pale parts of the chopped green onions, ginger, and red pepper. Continue to cook, tossing often, until green onions are softened and the bottom of skillet is starting to brown, about 1 minute.
Add cabbage, udon noodles, mirin, and soy sauce. Cook, tossing constantly, until noodles are coated in sauce (be sure to scrape bottom of skillet to dissolve any browned bits), about 45 seconds.
Remove from heat. Top with dark-green parts of scallions and serve.