In a large skillet over medium heat, add the ground beef and cook until browned, breaking it apart with a spatula, about 5-7 minutes. Drain excess fat if necessary.
Add the minced garlic, soy sauce, fish sauce, sugar, sesame oil, and black pepper to the skillet. Stir well and cook for another 2-3 minutes until the flavors meld together. Remove from heat and let cool slightly.
Lay out the lettuce leaves on a clean surface. Place a spoonful of the beef mixture in the center of each leaf. Top with shredded carrots and cucumber.
Carefully roll the lettuce leaf around the filling, tucking in the sides as you go to create a wrap.
Serve the rolls with lime wedges on the side for squeezing over the top.