In a large skillet or wok over medium-high heat, warm the sesame oil and 1 tablespoon of neutral oil. When it's hot and shimmering, add the chicken pieces and let them cook for 1 minute without stirring.

Stir in 2 tablespoons of soy sauce, 1 tablespoon of butter, fresh lemon juice, salt, and pepper. Cook until the chicken turns no longer pink, about 1 to 3 minutes, stirring gently only once or twice.

Carefully transfer the cooked chicken to a medium bowl and set it aside. Wipe the skillet clean with a paper towel to use for the next step.

Heat 1 teaspoon of neutral oil in the skillet over medium-high heat. Add the diced onion and frozen mixed vegetables, sautéing until the onions become almost translucent, roughly 3 minutes.

Move the sautéed vegetables to one side of the skillet. Crack the eggs directly into the skillet and scramble them with a spatula until they're cooked through.

Add the cooked rice and 3 tablespoons of butter to the skillet. Stir frequently for about 5 minutes or until the rice is heated through. Then stir in the remaining 3 tablespoons of soy sauce and cook for an additional minute.

Gently fold in the reserved chicken, adjusting salt and pepper to taste. Cook until everything is well combined and heated through. Portion into bowls, garnish with sesame seeds if you wish, and serve warm.