In a medium bowl, whisk together the low-sodium soy sauce, rice vinegar, honey, Sichuan chili crisp, grated garlic, grated ginger, and 2-3 tablespoons of water. Adjust the amount of water to achieve your desired sauce consistency. Set aside.
In a large skillet or wok, heat 1 tablespoon of neutral cooking oil over medium-high heat. Add the sliced chicken and stir-fry for about 5-7 minutes, or until the chicken is cooked through and lightly browned. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining tablespoon of cooking oil. Add the white parts of the scallions, shredded cabbage, and carrot. Stir-fry for 3-4 minutes, or until the vegetables are tender but still crisp.
Return the cooked chicken to the skillet with the vegetables. Pour the prepared sauce over the chicken and vegetables, tossing everything together to coat evenly. Cook for an additional 2-3 minutes, allowing the sauce to heat through.
Add the cooked udon noodles to the skillet, gently tossing them with the chicken and vegetables until everything is well combined and heated through. If the noodles seem sticky, add a splash of water to help loosen them.
Remove the skillet from heat. Serve the Ginger Chicken Udon Noodles hot, garnished with the green parts of the scallions, a sprinkle of toasted sesame seeds, and an extra drizzle of Sichuan chili crisp if you like it spicy.