Heat the vegetable oil in a large skillet or wok over medium heat.
Add the diced onion and sauté for about 2-3 minutes, until translucent.
Stir in the minced garlic and chopped carrot, cooking for another 2-3 minutes until the carrot is tender.
Add the chopped kimchi to the skillet and continue to stir-fry for 2 minutes to combine the flavors.
Increase the heat to high and add the cooked rice, breaking apart any clumps. Stir well to mix everything evenly.
Drizzle the soy sauce and mix in the gochujang, allowing the rice to absorb the flavors. Stir in the frozen peas.
If using eggs, push the rice mixture to one side of the skillet. Crack the eggs into the cleared space and scramble until just set, then mix with the rice.
Season with salt and pepper to taste, then remove from heat.
Serve the fried rice hot, garnished with sliced green onions, extra kimchi, and a sprinkle of sesame seeds.