Whisk the broth, soy sauce, sugar, cornstarch, and sesame oil in a small bowl until the sugar and cornstarch are completely dissolved.

Bring water to a boil in a Dutch oven. Cook the noodles and salt until tender, 2 to 3 minutes, then drain thoroughly in a colander. Toss the noodles with the scallions and more salt in a large bowl to combine.

Adjust an oven rack to the middle position and heat the oven to 200 degrees. Set a wire rack inside a rimmed baking sheet. Heat oil in a 12-inch nonstick skillet over medium heat until shimmering. Spread the noodles evenly into the skillet and press firmly into a flat cake with a spatula. Cook without moving until the bottom is crisp and deeply golden brown, 5 to 8 minutes.

Slide the noodle cake onto a large plate. Add more oil to the skillet and swirl to coat. Invert the noodle cake onto a second plate, then slide it, browned side up, back into the skillet. Cook until the second side is golden brown and crisp, 5 to 8 minutes. Slide the finished cake onto the prepared wire rack and keep it warm in the oven.

Combine a small amount of oil with the ginger and garlic in a small bowl. Add the remaining oil to the empty skillet and return to high heat until the oil is just smoking. Add the bok choy stalks and red bell pepper and cook until they are lightly browned and crisp-tender, 2 to 3 minutes.

Clear the center of the skillet, add the ginger mixture, and cook, mashing the aromatics into the pan, until fragrant. Stir the ginger mixture into the vegetables, then add the bok choy greens and cook until they just begin to wilt. Whisk the sauce to recombine, then add it to the skillet. Cook, stirring constantly, until the sauce has thickened and glazed the vegetables. Serve immediately over the warm noodle cake.