Fresh Chinese noodles are often kept in the produce section of the grocery store. If you can’t find them, substitute an equal amount of fresh Italian spaghetti. The chicken is easier to slice if you place it in the freezer for 20 minutes before slicing.
1. For the noodle cake: Bring 6 quarts water to a boil over high heat in a large pot. Add the noodles and salt to the boiling water; cook the noodles until tender, 2 to 3 minutes (do not overcook). Drain thoroughly through a colander. Add the scallions to the noodles in the colander and toss to combine; set aside.
2. For the sauce: Meanwhile, whisk together the sauce ingredients; set aside.
3. For the chicken and vegetables: Toss the chicken with the soy sauce and sherry; set aside. In a large bowl, whisk together the sesame oil, cornstarch, and flour; set aside. In a small bowl, mix together the ginger, garlic, and 1 teaspoon peanut oil; set aside.
4. While the chicken marinates, cook the noodle cake. Heat 2 tablespoons of the vegetable oil in a 12-inch nonstick skillet over medium heat until shimmering. Spread the noodles evenly across the bottom of the skillet and press with a spatula to flatten into a cake. Cook until crisp and golden brown, 5 to 8 minutes.
5. Slide the noodle cake onto a large plate. Add the remaining 2 tablespoons vegetable oil to the skillet; swirl to coat. Invert the noodle cake onto a second plate and slide it, browned side up, back into the skillet. Cook until golden brown on the second side, about 5 to 8 minutes.
6. Slide the noodle cake onto a cutting board and let sit for at least 5 minutes before slicing into wedges and serving. (The noodle cake can be transferred to a wire rack set over a baking sheet and kept warm in a 200-degree oven for up to 20 minutes).
7. Stir the marinated chicken into the sesame oil-cornstarch mixture until uniformly combined. Heat 2 teaspoons peanut oil in a 12-inch nonstick skillet over high heat until smoking; add half of the chicken to the skillet in a flat, even layer. Cook, without stirring, but gently separating the pieces, until golden brown on the first side, about 1 minute; turn the chicken pieces and cook until lightly browned on the second side, about 30 seconds. Transfer the chicken to a clean bowl. Repeat with an additional 2 teaspoons peanut oil and the remaining chicken; cover to keep warm.
8. Add the remaining 1 tablespoon peanut oil to the now-empty skillet and return to high heat until just smoking. Add the bok choy stalks and red bell pepper and cook until lightly browned, 2 to 3 minutes.
9. Push the vegetables to the sides of the skillet to clear the center; add the garlic-ginger mixture to the clearing and cook, mashing the mixture with a spoon, until fragrant, 15 to 20 seconds, then stir the mixture into the stalks and continue to cook until the stalks are tender-crisp, about 30 seconds longer. Stir in the bok choy greens and cook until beginning to wilt, about 30 seconds.
10. Return the chicken to the skillet. Whisk the sauce to recombine, then add to the skillet and cook, tossing constantly, until the sauce is thickened and the chicken is heated through, about 30 seconds. Transfer to a serving platter and serve immediately.