Add the ground beef and cook, breaking it apart with a wooden spoon, until browned and cooked through.
Drain excess grease from the skillet.
Add the sesame oil, garlic, and ginger. Cook for 1 minute.
Stir in the soy sauce, brown sugar, rice vinegar, and sriracha.
Reduce the heat to medium and cook for 3 to 4 minutes, stirring occasionally, until the beef is coated and the sauce has slightly thickened.
Remove the skillet from the heat.
Crack the eggs into a mixing bowl. Add the milk, salt, and black pepper. Beat with a wire whisk until blended.
Heat the olive oil in a medium skillet over medium heat.
Add the egg mixture. Cook, gently stirring, until the eggs are softly scrambled and set.
Warm the tortillas according to package directions until flexible.
Place about ⅙ of the scrambled eggs down the center of each tortilla.
Top each with about ⅙ of the Korean beef. Sprinkle each burrito with Monterey Jack cheese, shredded carrots, and green onions.
Fold the sides of each tortilla inward. Roll each tortilla tightly around the filling to form a burrito.
Notes
Tips and Tricks
Don't Overcook the Eggs - Remove the eggs from the heat when they're just set. They'll continue cooking slightly, especially if you're warming the finished burritos.Drain the Beef Well - Excess grease can make the tortillas soggy. This is especially important if you're preparing the burritos ahead of time.Keep the Filling Away From the Edges - Leave enough room on each side of the tortilla for folding. These burritos have plenty of filling, so don't spread it too far.Watch the Sauce - The brown sugar can cause the sauce to thicken quickly. Keep the heat at medium and stir occasionally so it doesn't burn.Toast the Finished Burritos - For a lightly crisp tortilla, place the wrapped burritos seam-side down in a dry skillet over medium heat for a minute or two per side.
Storing and Reheating
Refrigerator - Allow the filling to cool before wrapping the burritos. Store in airtight containers or wrap individually in foil. Refrigerate for up to 3 to 4 days.Freezer - Wrap each cooled burrito tightly in plastic wrap or foil, then place in a freezer-safe bag. Freeze for up to 2 to 3 months for best quality. For better texture, leave out the carrots and green onions before freezing.Microwave - Remove any foil and wrap the burrito in a damp paper towel. Microwave for 1 to 2 minutes if refrigerated, turning halfway through. Frozen burritos will take longer. Heat until the center reaches 165°F.Oven - Wrap the burritos in foil and heat at 350°F for approximately 15 to 20 minutes if refrigerated, or longer if frozen, until heated through.