Mix gochujang, soy sauce, sesame oil, and minced garlic in a small bowl. Set aside.

Heat 1 tbsp oil in a large skillet over high heat. Add onion and carrot; cook, stirring often, until softened and browned, 3 to 4 minutes. Stir in chicken if using and cook 1 minute.

Add cold rice and break up clumps. Spread into an even layer and cook undisturbed 2 minutes. Toss and repeat until crispy in spots, about 2 more minutes. Pour sauce over rice and toss to coat, cooking 1 minute.

Wipe skillet clean, add a little oil, and fry eggs to desired doneness. Serve rice topped with eggs, scallions, and sesame seeds.