1. Whisk the chicken broth, soy sauce, sherry, chili sauce, sugar, cornstarch, 2 teaspoons of the roasted sesame seeds, 1 teaspoon of the sesame oil, and 1 teaspoon of the garlic together in a small bowl until combined. Combine the remaining 2 teaspoons garlic, the ginger, and 1 teaspoon peanut oil in a small bowl.
2. Heat 2 teaspoons peanut oil in a 12-inch nonstick skillet over high heat until smoking; add half of the chicken in a flat, even layer. Cook until golden brown on the first side, about 1 minute; turn the chicken pieces and cook until lightly browned on the second side, about 30 seconds. Transfer the chicken to a clean bowl. Repeat with an additionally 2 teaspoons peanut oil and the remaining chicken.
3. Add 1 tablespoon peanut oil to the now empty skillet; heat until just smoking. Add the green beens and cook stirring occasionally, for 1 minute; add the mushrooms and stir-fry until the mushrooms are lightly browned and softened, about 3 minutes. Push the vegetables to the sides of the skillet to clear the center; add the garlic-ginger mixture to the clearing and cook, mashing the mixture with a spoon, until fragrant and gold brown, 30 to 45 seconds, then stir the mixture into the beens and mushrooms. Continue to stir-fry until the beans are tender-crisp, about 30 seconds longer. Return the chicken to the skillet. Whisk the sauce to recombine, then add to the skillet; reduce the heat to medium and cook, stirring constantly, until the sauce is thicken and the chicken is cooked through, about 30 seconds. Transfer to a serving platter, drizzle with the remaining 1 teaspoon sesame oil , and sprinkle with the remaining 1 teaspoon sesame seeds.