Heat 1 tablespoon of oil in a wok over high heat until it smokes. Add broccoli and fresh mushrooms. Stir in straw mushrooms, bamboo shoots, and water chestnuts. Stir-fry until veggies are crisp-tender, about 5 minutes. Transfer to a bowl and wipe out the wok.
Return wok to high heat. Add remaining 1 tablespoon oil, then stir in garlic and ginger. Quickly add sliced chicken. Stir-fry until the chicken is no longer pink, about 3–4 minutes.
Pour in chicken broth, soy sauce, and oyster sauce. Bring to a simmer.
Stir in the cornstarch slurry until the sauce thickens to a silky consistency, about 1–2 minutes.
Return vegetables to the wok. Toss everything together so each piece is coated in the luscious sauce. Season with salt and pepper.
Garnish with green onions or sesame seeds if desired. Serve hot over steamed rice or noodles.