Instructions

Whisk ⅓ cup ketchup, 3 tablespoons soy sauce, 1 teaspoon sugar, ¾ teaspoon pepper, ¾ teaspoon garlic powder, and ½ teaspoon kosher salt together in small bowl.
Heat 1 tablespoon vegetable oil in 12-inch skillet over medium-high heat until shimmering. Add 12 ounces chicken thighs, trimmed and cut into ½-inch pieces, and remaining ¾ teaspoon salt and cook, without stirring, until browned on bottom, about 2 minutes. Add ½ chopped onion and cook until chicken is cooked through and onion is beginning to soften, 2 to 4 minutes.
Stir in 4 cups cooked short-grain rice. Add ketchup mixture and cook, stirring constantly, until sauce is fully absorbed and coats rice evenly, about 1 minute. Remove pan from heat. Scoop one-quarter of rice mixture into small heatproof bowl, packing firmly. Invert 1 serving plate on top of bowl of rice, then flip and remove bowl so rice is mounded on plate. Repeat with remaining rice mixture, dividing among 3 additional plates. 
Make one omelet at a time. Whisk 1 egg with ½ tablespoon milk and ½ tablespoon mayonnaise in small bowl. Heat 8-inch nonstick skillet over medium heat until hot, about 2 minutes. Add 1 tablespoon unsalted butter and reduce heat to low. Swirl pan to coat with butter and, before butter fully melts, pour in egg mixture. Cook for 5 seconds without stirring, then use chopsticks to stir from center in swirling motion to scramble. Cook until egg is mostly cooked through but top is slightly runny, about 1 to 2 minutes. Carefully slide egg over first plate of rice. Repeat with remaining 3 eggs, 1½ tablespoons milk, 1½ tablespoons mayonnaise, and 3 tablespoons butter. Sprinkle eggs with 1 teaspoon minced fresh parsley and serve