Teriyaki Sauce

1. To make the sauce, heat oil in a wok or large skillet over medium heat. Add garlic, ginger and scallions and cook until fragrant, about 30 seconds. Add soy sauce, mirin and brown sugar and bring to a simmer over medium-high. Cook until reduced slightly, about 1 minute. Stir together cornstarch and 1 tablespoon water in a small bowl until smooth then add to the sauce. Cook until thickened, about 1 minute. Add water until sauce is pourable, but still thick. Transfer mixture to a small bowl. (There's no need to clean the wok or skillet after making the sauce)

Udon and Vegetables

1. Heat 2 tablespoons of the oil in the wok or skillet over medium-high and add the broccoli. Cook, stirring frequently, until bright green, about 30 seconds. Add cabbage and cook, stirring frequently until the cabbage just starts to wilt, another 30 seconds. Add remaining 1 tablespoon oil and the carrots and cook, stirring constantly until the vegetables are just starting to become tender, about 1 minute (adjust heat to avoid burning).
2. Move vegetables to the edge of the wok or skillet and add the noodles to the middle along with 1/4 cup water. Cover and cook for 2 minutes (if your wok doesn't have a lid, use a sheet tray). Remove lid and gently stir to separate the noodles, taking care not to break them up.
3. Add reserved sauce and gently toss until the noodles and vegetables are well coated. You may need to add some water to loosen the sauce and noodles. Serve garnished with sesame seeds.