In a medium saucepan, bring the rice and 3 cups water to a boil; lower the heat and simmer, covered, until the water is absorbed, about 20 minutes. Remove from the heat and let stand for 5 minutes.
In a medium bowl, whisk together half of the garlic, the soy sauce, sugar and 1/2 tablespoon cornstarch; season with salt and pepper. Add the chicken and toss; let stand for 20 minutes. In a small bowl, whisk together 1/2 cup water and the remaining 1 tablespoon cornstarch.
In a large nonstick skillet, heat 1 tablespoon olive oil and the remaining garlic over medium-high heat. Add the broccoli stalks, season with salt and pepper and cook for 1 minute. Add 1/4 cup water and the broccoli florets, cover and steam until crisp-tender, 3 to 4 minutes; transfer the broccoli to a plate.
Add the remaining 2 tablespoons olive oil to the skillet and heat until hot. Add the chicken mixture and scallion whites and cook, stirring, until the chicken is just cooked through, about 2 minutes. Stir in the broccoli, then stir in the remaining cornstarch mixture. Cook until the liquid thickens, about 30 seconds. Remove from the heat and stir in the scallion greens. Serve with the rice.