In a small bowl, combine the soy sauce, mirin, sake, hoisin, water, sesame oil, sugar or honey, and rice vinegar if using. Set aside.

Toss the sliced chicken with 1 teaspoon soy sauce, 1 teaspoon sake, potato starch, and a little black pepper. Let sit for 5–10 minutes while preparing the vegetables.

Cook or loosen the 14 oz udon according to the package directions. Stop just before fully cooked, since the noodles will finish in the skillet. Drain and set aside.

Heat 1 tablespoon oil in a large skillet or wok over medium-high to high heat. Add the chicken in a single layer and leave undisturbed for about 1 minute to develop some browning. Stir-fry for another 2–3 minutes, until almost cooked through. Transfer to a plate.

Add a little more oil if necessary. Add the mushrooms and cook for about 2 minutes, allowing them to brown before stirring too much.

Add the onion and bell pepper. Stir-fry for 2–3 minutes, until lightly browned but still slightly crisp. Add the green onion whites, garlic, and ginger and cook for another 30 seconds.

Return the chicken to the skillet and add the udon noodles. Pour the sauce around the edges of the hot pan. Toss everything together over high heat for 1–2 minutes, until the chicken is fully cooked, the noodles are hot, and the sauce has reduced into a glossy coating.

If the noodles become too sticky or the sauce reduces too quickly, add 1–2 tablespoons of water and toss again.

Remove from the heat and stir in most of the green onion greens. Taste before adding additional soy sauce. Garnish with the remaining green onions and, if desired, sesame seeds, chili crisp, or shichimi togarashi.