Heat vegetable oil in a large skillet or wok over medium-high heat.
Add the finely chopped onion and diced carrot to the skillet and sauté for 3-4 minutes until softened.
Add the minced garlic and cook for 30 seconds until fragrant.
Add the lean ground beef to the skillet and cook, breaking it apart with a spatula, until fully browned and cooked through, about 5-6 minutes.
Season the beef mixture with salt and freshly ground black pepper to taste.
Push the beef and vegetable mixture to one side of the skillet. Pour the beaten eggs into the empty side and scramble until just set, about 1-2 minutes.
Mix the scrambled eggs into the beef and vegetables.
Add the chilled cooked rice to the skillet. Stir well to combine all ingredients evenly.
In a small bowl, whisk together the gochujang, soy sauce, and sesame oil.
Pour the sauce over the rice mixture and stir thoroughly to coat everything evenly.
Add the thawed peas and cook for an additional 2 minutes, stirring frequently until heated through.
Remove the skillet from heat and stir in the sliced green onions.
Transfer the Korean beef fried rice to serving plates and sprinkle with toasted sesame seeds before serving.