Add the chicken and broccoli to a large bowl and set aside.
In a pot on medium heat, add the teriyaki sauce, carrots, lemon zest and juice, ginger, brown sugar, red chile flakes, scallions, garlic and some black pepper. Cook, stirring, until the mixture bubbles and the sugar is melted, about 5 minutes. Pour the hot sauce over the chicken and broccoli and toss to coat.
Scoop rice evenly into 4 bowls and top with some of the chicken and broccoli mixture, an egg and a sprinkle of French-fried onions.