Whisk together broth, soy sauce, and cornstarch in small bowl until smooth.
Heat a large skillet or wok over medium-high heat until drop of water sizzles in pan. Add 2 teaspoons oil and swirl to coat pan. Add chicken, sprinkle with salt and pepper, and cook, without stirring, 1 minute. Then stir-fry until chicken is cooked through, about 3 minutes. Transfer chicken to a plate.
Heat remaining 2 teaspoons oil in same pan. Add garlic and ginger and stir-fry until fragrant, 30 seconds. Add asparagus, bell pepper, and scallions; stir-fry until vegetables are crisp-tender, about 3 minutes. Return chicken to pan. Whisk broth mixture again and add to pan. Cook, stirring constantly, until mixture bubbles and thickens, about 1 minute. Remove from heat and stir in basil. Sprinkle with sesame seeds.