Heat 1 tablespoon oil in a large saute pan over medium-high heat. Add the bulgogi and cook until done, about 5 minutes. Transfer the bulgogi with its juices to a bowl and set aside.
Add butter to the same pan and heat until melted. Add the chopped onion and garlic, sauteing until the onion has softened, about 3-4 minutes. Add the chopped kimchi and cook until the kimchi has softened, about 5-6 minutes. Add the bulgogi back into the pan with the juices and stir to combine.
Add the cooked rice into the pan and stir well together. Add the kimchi juices, reduce heat to medium and continue to stir until every grain of rice is coated with the kimchi mixture. Taste and adjust seasoning, adding soy sauce or salt if needed. Portion the kimchi fried rice onto two plates.
Meanwhile, heat 1 teaspoon oil in a nonstick pan over medium heat. Whisk together the eggs with the salt. Pour half of the egg mixture into the pan and swirl the pan around to create an even layer of the egg. Once the eggs have set, flip the omelet over and cook on the other side. Remove from the pan and carefully lay on top of the kimchi fried rice. Repeat with the remaining egg. Serve with ketchup if desired.
Notes
You can find marinated bulgogi in Korean supermarkets or large Asian markets. You can also prepare the marinade at home and combine it with thinly sliced ribeye for homemade bulgogi.Season the fried rice with soy sauce and salt if needed. Storebought kimchi and bulgogi can vary in levels of saltiness so adjust the seasoning for the fried rice accordingly.