Break up any clumps in the cold rice with a fork or your fingers before you start cooking.
Heat 3 tablespoons of the oil in a large nonstick or carbon steel pan over medium-high heat. Add the onion, carrot, and broccoli and cook 2 to 3 minutes, until the onion edges turn translucent.
Add the ketchup, soy sauce, garlic, and a generous amount of black pepper, along with the butter if using. Stir together and cook for 1 minute; the ketchup should thicken and darken to a deeper brick color.
Add the cold rice and toss for 2 minutes, until every grain is evenly coated and the rice sizzles.
Add the frozen peas and toss for 1 minute, until warmed through and still bright green. Taste and season with salt.
Optional crispy bottom: add a bit more oil to the pan, press the rice into a flat, even layer, and leave it undisturbed until the bottom layer sets into a crust and the sizzle changes. Watch it closely.
Garnish with chopped scallions and sesame seeds and serve hot.