Bring rice and water to a boil in a saucepan over high heat. Reduce heat to medium-low, cover, and simmer until rice is tender, and liquid has been absorbed, 20 to 25 minutes.
Combine soy sauce, brown sugar, and corn starch in a small bowl; stir until smooth. Mix ginger, garlic, and red pepper into sauce; coat chicken with marinade and refrigerate for at least 15 minutes.
Heat 1 tablespoon sesame oil in a large skillet over medium-high heat. Cook and stir bell pepper, water chestnuts, broccoli, carrots, and onion until just tender, about 5 minutes. Remove vegetables from skillet and keep warm.
Remove chicken from marinade, reserving liquid. Heat 1 tablespoon sesame oil in skillet over medium-high heat. Cook and stir chicken until slightly pink on the inside, about 2 minutes per side; return vegetables and reserved marinade to skillet. Bring to a boil; cook and stir until chicken is no longer pink in the middle and vegetables are tender, 5 to 7 minutes. Serve over rice.
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