Prep ingredients: Bring a large pot of salted water to a boil. Trim ends from scallions and thinly slice on an angle; reserve 2 tablespoons scallion greens in a small bowl for step 6. Cut bok choy lengthwise into 1-inch wedges, and rinse well to remove any grit.
Pound chicken: Pat chicken dry, and using a meat mallet or heavy skillet, pound to an even ½-inch thickness. Rub with 1 teaspoon sesame oil and season with ¼ teaspoon each salt and pepper. Let sit until step 5.
Blanch bok choy: Add bok choy to boiling water and cook until wilted, 1–2 minutes. Using a slotted spoon, transfer to a colander to drain. Shake out excess water from bok choy. Return water in pot to a boil.
Cook noodles: Add ⅔ of the noodles to pot (save rest for your own use). Cook, stirring, until tender, 6–7 minutes. Reserve ¼ cup cooking water, then drain and rinse noodles with lukewarm water; return to pot. Add tamari, 1 tablespoon butter, ½ teaspoon sesame oil, and scallion whites. Stir in 2 tablespoons cooking water. Remove from heat, and cover to keep warm.
Cook chicken: Meanwhile, heat 1 tablespoon oil in a medium skillet over medium-high until shimmering. Add bok choy and cook, turning once or twice, until browned in spots, 2–3 minutes. Transfer to a platter; cover to keep warm. Add 1 tablespoon oil to skillet. Add chicken and cook, turning once, until lightly browned and nearly cooked through, 2–3 minutes per side.
Finish & serve: Mix hoisin with ½ teaspoon sesame oil and remaining noodle cooking water and add to skillet. Cook on low, turning chicken until glazed with sauce, about 2 minutes. Serve noodles with chicken and bok choy. Drizzle hoisin sauce over chicken. Sprinkle all with sesame seeds and reserved scallion greens. Enjoy!