Whisk the sauce: in a small bowl combine the soy sauce, rice vinegar, honey, Sichuan chili crisp, grated garlic, grated ginger, and 2–3 tablespoons water until smooth; set aside.
Heat 1 tablespoon neutral oil in a large skillet over medium-high heat. Add the thinly sliced chicken in a single layer and cook, stirring occasionally, until browned and cooked through, about 4–6 minutes. Transfer chicken to a plate.
Add the remaining 1 tablespoon oil to the same skillet. Add shredded cabbage and julienned carrot and cook, stirring, until wilted and lightly browned, about 2–3 minutes.
Stir in the scallion whites and cook 1 minute more.
Add the cooked udon noodles to the skillet and toss to combine with the vegetables.
Return the chicken to the pan, add the scallion greens, pour in the prepared sauce, and toss everything together until the noodles are evenly coated and heated through, 1–2 minutes. If the pan seems dry, add 1–2 tablespoons water or a splash more soy sauce while tossing.
Serve immediately with extra chili crisp, toasted sesame seeds, and additional scallion greens if desired.
