Pat the chicken thighs dry with paper towels, then season on both sides with the salt and black pepper.

Crack the egg into a shallow bowl and beat until smooth. Place the flour on a large plate in an even layer.

Dip each chicken thigh into the beaten egg, letting the excess drip off, then coat thoroughly in the flour, pressing it gently onto the skin and meat. Set the coated pieces aside on a plate.

Heat 2 tablespoons of the neutral oil in a large skillet over medium-high heat until a pinch of flour sizzles on contact.

Place the chicken thighs skin side down in the hot skillet and cook without moving them for 6 to 8 minutes, until the skin is deep golden and crisp.

Turn the chicken pieces over and cook for another 6 to 8 minutes, until the meat is cooked through and the juices run clear when pierced.

While the chicken cooks, bring a pot of salted water to a boil. Add the noodles and cook according to package directions, usually 4 to 7 minutes, until just tender. Reserve about 0.25 cup of the cooking water, then drain the noodles.

In a medium bowl, whisk together the soy sauce, honey, minced garlic, ginger, and sesame oil until the sauce is smooth.

Reduce the heat under the skillet with the chicken to medium. Pour the sauce into the skillet around the chicken, scraping up any browned bits from the bottom with a spatula. Simmer for 3 to 4 minutes, spooning the sauce over the chicken until it thickens slightly and coats the meat.

Add the drained noodles and the sliced bell pepper or broccoli directly to the skillet. Toss gently with tongs to coat everything in the sauce, adding 1 to 2 tablespoons of the reserved pasta water if the mixture seems too thick.

Cook for 1 to 2 minutes, just until the vegetables are slightly tender but still crisp and the noodles are glossy and evenly coated with the sauce. Serve hot straight from the skillet.