Rinse and dice all vegetables. Slice ham into small squares. Heat butter over medium-high heat in a large pan with deep sides (or a wok.) Add vegetables and sliced ham once the butter is melted. Season with salt and cook until vegetables are tender, about 5-7 minutes.

Add cooked rice to the vegetables and stir, breaking up any lumps of rice. Whisk ketchup and soy sauce together and mix well into the rice.

Use a small bowl and fill it with rice mixture. Place a plate on top of the bowl, and flip while holding the bowl against the plate. Carefully lift the bowl from the plate so that the rice stays in a nice circular shape. Repeat step 3 for each serving. (Or skip the bowl flip and add 1 ⅓ cup of rice to each plate.)

Make the omelet: In a small dish, add 2 eggs, season with salt, and scramble with a fork. In a small lightly greased pan, heat to medium-high temperature. Once hot, add scrambled eggs and rotate the pan so that egg spreads and covers the bottom of the pan. Once the bottom of the eggs are cooked and the top is slightly runny, turn off the heat.

Carefully transfer the omelet, runny side down onto the rice. The cooked side should be face up. Top with a drizzle of ketchup. Repeat step 4 for each serving.