Mix ketchup, memmi, mirin together in a small bowl.

Melt butter over medium-high heat in a frying pan. Stir-fry onion in the butter until translucent, about 5 minutes. Add chicken, vegetables, and 1/2 the ketchup mixture. Sprinkle with salt and pepper to taste. Stir in cooked rice and remaining ketchup mixture. Stir-fry until rice is coated and chicken is no longer pink in the center, 5 to 7 minutes. Divide fried rice mixture between 2 plates and mold each into compact, oblong shapes.

Whisk 2 eggs and 1 tablespoon milk for 1 omelet together in a small bowl. Heat a nonstick pan over medium heat.

Pour beaten egg mixture into the hot pan; let cook for 30 seconds. Lift the edges of the omelet so that the uncooked egg runs under the cooked edges and comes into contact with the hot skillet. Shake and tilt the skillet to move the uncooked egg. Repeat until the top is set, about 2 to 3 minutes. Lay the omelet on top of 1 fried rice mixture and tuck under. Repeat with remaining 2 eggs and 1 tablespoon milk for the second omelet; lay over remaining fried rice mixture and tuck.