In a small bowl, mix together the dark soy sauce, oyster sauce, honey, mirin, red chili flakes, and black pepper. Set aside.
Heat the avocado oil in a large pan over medium-high heat. Add the sliced onion and minced garlic, sauté for about a minute, then add the chicken pieces and sauté until golden brown.
Add the shredded cabbage, matchstick carrots, and sliced mushrooms to the pan. Sauté until tender but still crunchy.
Bring another pot of water to a boil. Cook the udon noodles according to package instructions, then drain and rinse under cold water.
Add the cooked udon noodles to the pan with the chicken and veggies. Pour the sauce over and toss everything together until well coated.
Garnish with sliced green onions and serve immediately.