Heat 2 teaspoons vegetable oil in a large lidded skillet or wok over medium-high heat until it shimmers. Add pepper and onion and cook, stirring occasionally, until crisp-tender, about 5 minutes. Transfer to a bowl.

Add another 2 teaspoons oil to the skillet and heat until it shimmers. Add chicken and cook, stirring occasionally, until no longer pink, about 5 minutes. Transfer to bowl with vegetables.

Add another 2 teaspoons oil and heat until it shimmers. Add chopped zucchini, carrots, cabbage, and snap peas; cook, stirring occasionally, 2 minutes. Add broth and cook, covered, until vegetables are crisp-tender, about 3 minutes more. Transfer to bowl.

Add remaining 3 teaspoons oil to the skillet and heat until it shimmers. Add garlic and ginger, cook and stir for 15 seconds. Add rice and cook, breaking up clumps and stirring occasionally for 2 minutes. Add bowl contents, soy sauce, sesame oil, green onions, and salt. Cook, stirring and tossing, until well mixed and heated through, about 5 minutes more.