Bring water to a boil in a saucepan and stir in rice according to directions. Cover, remove from heat, and let stand for 5 minutes.

Meanwhile, heat 2 tablespoons olive oil in a large skillet over medium heat. Add chicken and onion and cook until onion is soft and translucent and chicken is no longer pink inside, about 5 minutes.

Crack egg into a bowl, add milk, and beat to combine. Pour into the skillet with chicken. Season with salt and pepper. Cook and stir until egg is scrambled and set, about 5 minutes. Add cooked rice and cook for 5 more minutes. Drizzle remaining tablespoon olive oil on top and season with additional salt and pepper to taste.