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For the ketchup rice: Steam the rice, preferably a day ahead (it makes for a firmer, heartier fried rice). If you're tight on time and need to cook it day of, use a little less water so the rice is on the drier, firmer side. Let cool.
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To a small nonstick saute pan (5- to 6-inch), add the oil and bring up to medium heat. Season the chicken with salt, and once the oil is barely shimmering, add the chicken, stirring to coat with oil, cooking almost all the way through, about 2 to 3 minutes. Try not to brown the chicken at all so your meat stays tender. Add the carrots and onions and cook the mixture for 1 minute more.
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Add the rice, stirring to evenly disperse all of the ingredients. Make a well in the middle of the pan, clearing away the rice mixture so there's a hole and you can see the bottom of the pan. Add the ketchup to the center and let cook undisturbed for about 20 to 30 seconds, or until it stops bubbling. Mix the reduced ketchup into the rice. Remove the pan from the heat and add black pepper and salt to taste.
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Fresh ketchup rice tastes best, but it will keep for a few days, and reheats well too. If you're going to eat right away, I suggest plating the rice while it's still warm and then making the omelet. Pile the rice on a plate, and with a rubber spatula, form an omelet-shaped ovular mound about 1-inch high. Tip: Cover the rice with saran wrap and mold it using your hands (it's easier, and cleaner).
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Wipe out the saute pan with a paper towel (you'll use it for the omelet too).
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For the omelet: Crack eggs into a small bowl, adding milk and salt, and whisk together until well combined.
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To the nonstick pan, over medium-high heat, add the butter and let melt until it starts to bubble, swirling the pan to coat all the way to the edges. Working quickly, add the eggs, and stir them in the pan with wooden saibashi (Japanese cooking chopsticks) to form long curds, pulling the more cooked egg from the edges into the center, while still scrambling the eggs in the center of the pan as well. If you don't have saibashi, a small wooden spatula works in a pinch.
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When the eggs are just about half way cooked, or after about 60 to 90 seconds, there will be a nest of nearly cooked eggs on the bottom, with a combination of large curds and undercooked eggs on top. While the egg on the bottom of the pan starts to set, turn off the burner, and let the pan sit for about 1 minute. This will firm up the bottom and edges, enabling you to loosen and fold the egg omelet. (I'll warn you, this may take a few attempts-I went through a full dozen eggs in an effort to get this right.)
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Next, you're going to attempt to fold the omelet onto itself, sealing an edge while keeping a portion of the half-cooked eggs inside. Working toward the furthest edge of the pan, tilt the pan at a 30-degree angle away from you, and from the side closest to the handle, slowly fold up the edges of the omelet toward the center, by quarters, if not thirds, eventually working your way all the way to the outer edge. You can also slide the omelet onto itself about 3/4 of the way, and then knock the handle of the pan lightly with the heel of your wrist on your non-dominant hand. This will cause the omelet to jump a little, while sliding onto itself. I suggest watching a few videos on this technique before giving it a go. Finally, you'll have a rolled omelet curled up at the furthest edge of your pan. You'll need a little more heat to really seal that edge, so put only that far edge of the pan over medium-high heat for a few seconds.
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Plate immediately. To do so, you're going to invert the omelet on top of the rice mound. Sounds harder than it is, but what you'll do is reverse your hand position on the pan, taking an under-hand grip on handle with your thumb now furthest away from the pan, so you can more easily slide the omelet onto the plate. Gently slide, almost flipping the omelet on top of the rice, guiding it with the saibashi, so it doesn't fall off the rice mound, making sure it's stable before you take the pan away. Once you have it balanced on top of the rice, bring it to the table, and with a sharp knife, using only the tip, slice the omelet from end to end, lengthwise, unfurling the omelet as if it were a flower blooming.
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Garnish with ketchup, if you'd like.
