Mix together garlic, ginger, lemon juice, honey, and soy sauce in a small bowl.
Preheat oven to 450 degrees. Coat broccoli with the olive oil and bake until roasted and crisp, about 20 minutes. Remove from oven and sprinkle with salt.
In the meantime, heat a small amount of oil in a pan to a medium heat. Add chicken, basting with the remaining marinade slowly throughout, and cook until the chicken reaches an internal temperature of 165 degrees.
Remove the chicken and any remaining marinade and add the cabbage to the pan. Sprinkle with salt and cook until slightly soft and the color becomes more vibrant. Remove from heat.
Cut the chicken into thin slices (save any remaining marinade in a small bowl). Peel the eggs and cut each one in half.
Divide rice between two bowls. Top each bowl with half the chicken, cabbage, broccoli, one egg, green onion, sesame seeds, and any remaining marinade. Serve with soy sauce if desired.