Prepare vermicelli rice noodles according to package instructions. Drain well, rinse with cold water, and set aside.

Heat avocado oil in a large skillet over medium-high heat. Add garlic and shallot; cook until fragrant, about 1 minute. Add ground beef and cook, breaking it up, until browned. Drain excess fat.

Stir in the soy sauce, fish sauce, and brown sugar. Cook for 2-3 minutes until the sauce thickens. Remove from heat and stir in the fresh Thai basil and lime juice. Allow to cool slightly.

In a small bowl, whisk together all dipping sauce ingredients until the sugar is dissolved. Set aside.

Fill a shallow dish with warm water. Submerge one rice paper wrapper for 15-20 seconds until pliable. Lay it flat. On the bottom third, layer a lettuce leaf, a small amount of noodles, carrots, cucumber, and about 2-3 tablespoons of the beef filling.

Fold the bottom edge up over the filling, fold in the sides, and then roll tightly to the top. Repeat with remaining ingredients.

Serve the Thai Basil Beef Rolls immediately with the chili-lime dipping sauce.