Thinly slice the chicken and season it with salt and pepper. Slice the mushrooms, bell pepper, and carrots, and cut the broccoli into small florets.
Heat 1 tablespoon vegetable oil in a large skillet or wok over medium-high heat. Cook the chicken until no longer pink, 5 to 7 minutes, then transfer it to a plate.
Add the remaining tablespoon of oil to the same pan. Cook the garlic and ginger until fragrant, about 1 minute.
Add the mushrooms, snow peas, broccoli, bell pepper, and carrots. Stir-fry until tender-crisp, 5 to 7 minutes.
Return the chicken to the pan. Stir in the soy sauce, chicken broth, and oyster sauce and bring the mixture to a simmer.
Stir in the cornstarch slurry and cook until the sauce thickens, 2 to 3 minutes.
Garnish with green onions and sesame seeds if desired, and serve hot.