Cook the rice (it's best to make it a day ahead and let it sit in the fridge, but having it fresh won't be the end of the world). Prepare the sauce by stirring together the ketchup, hoisin sauce, and soy sauce in a small bowl. Set aside.
In a large skillet over medium heat, melt the butter. Cook the white parts of the onions until tender, about 4 minutes. Add the minced garlic and stir, cooking until aromatic.
Turn the heat to high and add the mixed vegetables, stirring constantly to prevent burning. After the veggies are tender, add the rice and the sauce, stirring constantly to prevent burning and making sure everything gets mixed well. After the rice is all warmed through (3-4 minutes), remove from the heat and turn off the burner.
In a small bowl, add two eggs (for one omelet). Beat well with a pinch of salt and a pinch of sugar. Pour a tablespoon of olive oil into a small skillet and set over medium heat. Add the eggs, swirling to cook gently. When the bottom has set but the top is still slightly runny, gently fold the omelet over itself in the pan so the residual heat gradually cooks the insides. Repeat for the omelets for the other servings.
Plate the rice in a mound, and top gently with the omelet. Garnish with extra ketchup and the green onion tops, and enjoy!