4 tablespoons peanut or canola oil, divided
1 large sweet or yellow onion, thinly sliced (about 2 cups)
2 cups broccoli florets, cut into small pieces
3 medium carrots, cut into matchsticks (1 1/2 cups)
2 medium zucchini, cut into 2-inch pieces (1 cup)
1 cup snow peas, trimmed and halved lengthwise
1/2 cup soybean or mung bean sprouts (about 2 ounces)
1 1/2 pounds boneless, skinless chicken breasts, cut into strips
1 teaspoon kosher salt
1/2 teaspoon freshly ground black pepper
1/2 cup plus 3 tablespoons low-sodium chicken broth, divided
1/2 cup reduced-sodium soy sauce or tamari
1/4 cup mirin or rice wine
3 tablespoons packed brown sugar
2 tablespoons honey
1 tablespoon grated or minced peeled fresh ginger
3 small cloves garlic, grated or minced (about 1 tablespoon)
2 tablespoons cornstarch
3 cups cooked rice
Heat 2 tablespoons of the oil in a 3 1/2- to 5 1/2-quart Dutch oven over medium-high heat until shimmering. Add the onions, broccoli, and carrots and sauté until the broccoli is a bright green and the carrots are still crisp but slightly tender, about 2 minutes.
Add the zucchini, snow peas, and sprouts and cook, stirring, until heated through, about 1 minute. Transfer to a large plate and set aside.
Sprinkle the chicken with the salt and pepper on both sides.
Heat the remaining 2 tablespoons of oil in the same Dutch oven over medium-high heat until shimmering. Add the chicken in a single layer and cook until browned on both sides, 3 to 4 minutes total.
Add 1/2 cup of the chicken broth, soy sauce, mirin, brown sugar, honey, ginger, and garlic. Stir and scrape up any flavorful browned bits from the bottom of the pan. Reduce the heat to low and cook just until the mixture comes to a gentle boil, 2 to 3 minutes.
Cover, reduce the heat to a simmer, and cook until the chicken is just cooked through and no longer pink, 5 to 6 minutes.
Whisk the cornstarch with the remaining 3 tablespoons chicken broth in a small bowl until fully combined. Stir the chicken and sauce in the Dutch oven, and drizzle in the cornstarch mixture. Increase the heat to medium and bring to a boil, stirring the liquid constantly, 1 to 2 minutes.
Add the rice and reserved vegetables and stir to combine. Cover and cook until the rice has begun to absorb the sauce, the vegetables are cooked to the desired doneness, and the chicken is soft and tender, about 3 to 5 minutes.