Bring water and rice 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.
Meanwhile, combine soy sauce, brown sugar, and cornstarch in a medium glass or ceramic bowl; stir until smooth. Stir in ginger, garlic, and red pepper flakes; add chicken and stir to coat. Cover and marinate in the refrigerator for at least 15 minutes.
Heat 1 tablespoon sesame oil in a wok or large skillet over medium-high heat. Add broccoli, onion, carrots, water chestnuts, and bell pepper; cook and stir until just tender, about 5 minutes. Transfer vegetables into a dish; set aside.
Heat remaining 1 tablespoon sesame oil in the same wok or skillet over medium-high heat. Add chicken, reserving marinade, and cook until just browned, about 2 minutes per side; stir in vegetables and reserved marinade. Bring to a boil; cook and stir until chicken is no longer pink in the center and vegetables are tender, 5 to 7 minutes. Serve over rice.