Heat 1 tablespoon of the avocado oil in a large 12-inch nonstick skillet over medium-high heat until shimmering.
Add the chopped chicken in a single layer and cook undisturbed a few minutes until golden, then stir and continue cooking until cooked through, about 4–6 minutes total. Transfer chicken to a plate and set aside.
Add the remaining 1 tablespoon avocado oil and the toasted sesame oil to the skillet and heat over medium-high.
Add the chopped shiitake mushrooms and sauté 3–5 minutes until they soften and begin to brown.
Stir in the sliced cabbage, minced garlic, and grated ginger; cook 1–2 minutes to combine.
Pour in the hoisin sauce, oyster sauce, coconut aminos (or soy sauce), and rice vinegar; stir to coat. Continue cooking, stirring occasionally, until the cabbage is softened to your liking, about 8–12 minutes.
While the cabbage cooks, heat a small nonstick pan or use the same skillet pushed to the side to scramble the eggs with a quick spray or a little oil; cook until just set, then transfer to the plate with the chicken.
When the cabbage is done, stir in the drained bamboo shoots, cooked chicken, and scrambled eggs. Add the green onion pieces and cook 1–2 minutes more, stirring constantly, until everything is heated through.
Serve the moo shu chicken mixture in bowls and enjoy, optionally with extra hoisin sauce on the side.
