Make the yakisoba: Place yakisoba noodles in a colander and rinse under room-temperature water. Using your hands, gently loosen and separate noodles. Drain well.
In a 12-inch high-sided nonstick skillet, heat 1 tablespoon of the oil over medium-high. Add noodles and spread in an even layer; cook undisturbed until golden and charred in spots, 3 minutes. Stir noodles once, then cook undisturbed until golden and charred in spots on the other side, 2 minutes longer. Transfer to a large plate.
Meanwhile, make the sauce: In a small bowl combine all of the ingredients and mix well.
To the skillet over medium, add the remaining 2 tablespoons oil, onion, carrot and bell pepper, and season with salt and pepper. Cook, stirring occasionally, until the vegetables are softened and charred in spots, 5 minutes. Stir in garlic until well combined, then add mushrooms and cook, stirring occasionally, until vegetables are tender and light golden, 3 to 5 minutes. Stir in kale in batches until wilted. Add the noodles, sauce and all but ¼ cup of the scallions, and cook, stirring occasionally, until sauce is absorbed, about 3 minutes. (Noodles should be nicely coated in the sauce but not soupy.) Season with salt and pepper.
Divide yakisoba among 4 plates and garnish with the remaining scallions. Serve warm