Heat a large skillet over medium-high heat. Add the olive oil and onion to the skillet and stir to coat. Cook the onion for approximately 2-3 minutes, or until translucent. Stir occasionally. Add the garlic to the skillet and cook for an additional one minute.
Stir in the carrots and the peas with the onion, cooking for approximately 5 minutes before adding the cooked rice, chopped kimchi, gochujang, riced cauliflower, soy sauce, and sesame oil. Reduce heat to medium, mix well and stir occasionally.
As the fried rice continues cooking, prepare your eggs, if serving.
Heat a medium skillet over medium-high heat allowing it to get nice and hot (you know it's hot when water droplets bounce right off the pan). Add the butter to the skillet and carefully crack the desired number of eggs in the skillet (careful not to overcrowd). Reduce heat to medium, medium-low and continue to cook until white had almost set, but yolk is still runny. Remove from heat.
Transfer desired amount of fried rice to individual serving bowls and top with additional kimchi, chopped green onion, and fried egg.