In a large skillet, heat the vegetable oil over medium heat. Add the minced garlic and ginger, sautéing for about 1 minute until fragrant.
Add the ground beef to the skillet, breaking it apart with a spatula. Cook until browned, about 5-7 minutes.
Stir in the soy sauce, oyster sauce, fish sauce, and brown sugar. Cook for an additional 2-3 minutes until the sauce is well combined and the beef is fully cooked.
Remove the skillet from heat and stir in the fresh Thai basil leaves until wilted.
To assemble the rolls, lay a lettuce leaf flat on a plate. Spoon a portion of the beef mixture onto the leaf, then top with shredded carrots and sliced cucumbers.
Drizzle with lime juice and roll the lettuce leaf around the filling, securing it as you go. Repeat with remaining ingredients.
Serve immediately with chili sauce on the side for dipping, if desired.