How to Make It
In a large pot of salted boiling water, cook the noodles until just tender, about 6 minutes. Drain and rinse under cold water. Drain the noodles thoroughly and transfer to a rimmed baking sheet. Toss the noodles with 1 tablespoon of the soy sauce and 2 teaspoons of the vegetable oil and spread in an even layer.
In a small bowl, whisk the sesame oil with the crushed white sesame seeds, sugar and the remaining 2 tablespoons of soy sauce; season with salt.
In a small bowl, beat the egg with 1 teaspoon of water and a pinch of salt. In a small nonstick skillet, heat 1 teaspoon of the vegetable oil. Add the egg and cook over moderate heat, swirling the pan to coat the bottom but not stirring, until the egg is just set, 2 minutes. Using a spatula, carefully flip the omelet and cook until just set, about 15 seconds more. Carefully slide onto a work surface and gently roll into a log, then slice into 1/4-inch-thick strips.
In a large wok, heat 1 tablespoon of the vegetable oil. Add the shrimp, garlic and a generous pinch of salt and stir-fry over moderately high heat until the shrimp begin to turn pink, about 2 minutes. Add the mirin and continue cooking until the shrimp are white throughout and the liquid has reduced, about 2 minutes more. Scrape the shrimp into a medium bowl. Wipe out the wok.
Heat the remaining 1 tablespoon of vegetable oil in the wok. Add the red onion and carrots and stir-fry over moderately high heat until the carrots are almost tender, about 4 minutes. Add the Broccolini, Swiss chard stems and a generous pinch of salt and stir-fry until all the vegetables are crisp-tender, about 5 minutes. Add the Swiss chard leaves, oyster mushrooms, sugar snaps and English peas and stir-fry until the mushrooms and peas are tender, about 2 minutes more.
Add the noodles to the wok in batches, stirring well after each addition. Add the sauce and stir-fry until the noodles are warmed through, 3 minutes. Add the cooked shrimp and their juices to the wok along with half of the sliced egg and toss gently. Transfer the jap chae to bowls; garnish with the remaining egg, the scallion and black sesame seeds.
Korean-style vermicelli (sweet potato starch noodles) is available at Asian markets and on amazon.com.