Whisk the broth, soy sauce, sherry, oyster sauce, sugar, cornstarch, and red pepper flakes in a small bowl until the cornstarch is dissolved.

Cook the noodles in a large pot of boiling salted water until just tender, then drain and rinse under cold water. Pat the noodles dry with a clean kitchen towel—removing excess moisture is key for a truly crispy exterior.

Toss the noodles with the scallions and oil. In a 10-inch nonstick skillet, heat oil over medium-high heat until shimmering. Spread the noodles into the skillet, pressing them down with a spatula to form an even cake, and cook until the bottom is golden brown and crisp, 5 to 7 minutes.

Slide the noodle cake onto a large plate, invert it onto a second plate, and then slide it back into the skillet with more oil. Cook the second side until golden and crisp, about 5 minutes, then transfer the cake to a cutting board and tent loosely with foil.

Wipe out the skillet and heat the remaining oil over medium-high heat. Add the ginger and garlic and cook until fragrant.

Add the bok choy stalks and bell peppers and stir-fry until the vegetables are tender-crisp, 3 to 4 minutes.

Whisk the sauce to recombine, then add it to the skillet along with the bok choy greens. Simmer until the sauce has thickened and the greens have wilted, about 2 minutes.

Cut the noodle cake into wedges and serve immediately with the hot stir-fry spooned over the top.