Prep. Pat the chicken dry with paper towels. Season both sides with the salt and pepper. In a small bowl, stir the soy sauce, mirin, brown sugar, and water until the sugar dissolves.
Sear the chicken. Heat 1 teaspoon oil in a large skillet over medium-high heat. Add the chicken and cook without moving it for 4 to 5 minutes, until deep golden. Flip and cook 3 to 4 minutes more.
Glaze. Lower the heat to medium. Pour in the sauce and cook for 1 to 2 minutes, turning the chicken, until the sauce thickens and coats it. If you use the cornstarch slurry, stir it in for the last 30 seconds. The chicken is done at 165°F (74°C) inside. Watch the pan closely, because the sugar can burn fast.
Rest and slice. Move the chicken to a cutting board and rest for 3 minutes. Slice into thick pieces. Leave the sauce in the pan, adding 1 tablespoon of water if it is too thick to drizzle.
Fry the eggs. Wipe the pan, or use a small nonstick pan. Heat 1 teaspoon oil over medium-low heat. Crack in the eggs and cook for 2 to 3 minutes, covering for the last 30 to 60 seconds so the whites set and the yolks stay runny.
Assemble. Divide the rice between two bowls. Top with the sliced chicken, then a fried egg. Drizzle with the pan sauce, then add the green onion and sesame seeds.