1. Cook the rice according to the package directions.
2. Meanwhile, in a bowl, combine the chicken broth, wine, soy sauce, sugar, sesame oil, cornstarch, and ½ teaspoon salt, stirring until the cornstarch dissolves.
3. Heat 1 tablespoon of the olive oil in a saucepan over medium-high heat for 30 seconds. Add the scallions and ginger and stir-fry until fragrant, about 15 seconds. Add the cornstarch mixture and bring to a boil. Cook, stirring constantly to prevent lumps, until sauce thickens slightly, about 1½ minutes. Pour into a bowl; set aside.
4. Heat a wok or large skillet over high heat. Add 2 tablespoons olive oil and heat for 30 seconds. Add the chicken and stir-fry until cooked through, 3 to 4 minutes. Transfer to a plate.
5. Wipe out the pan. Reduce heat to medium-high. Add the remaining ½ tablespoon oil and heat for 30 seconds. Add the carrots and ⅓ cup water, cover partially, and cook, stirring occasionally, until almost tender, about 2 minutes.
6.. Add the bok choy stems, cover, and cook for 2 minutes. Add the bok choy leaves and peas and toss gently. Add the sauce and increase heat to high. When it starts to bubble, add the chicken. Stir-fry until warmed through, about 1 minute. Serve immediately over the rice.