Heat sesame oil in a large skillet over medium-high heat. Add ground beef and cook until browned. Drain excess grease. Add garlic and ginger, cooking for 1 minute until fragrant. Stir in soy sauce, fish sauce, and brown sugar. Cook for 2-3 minutes until the sauce thickens. Remove from heat and stir in chopped Thai basil and shredded carrots. Set aside to cool.

Cook vermicelli noodles according to package directions; drain and rinse with cold water. Fill a shallow dish with warm water. Arrange all prepared ingredients (beef, noodles, lettuce, herbs, cucumber) on a clean work surface.

Submerge one rice paper sheet in warm water for 15-20 seconds until pliable. Lay it flat. In the bottom third, layer lettuce, noodles, herbs, cucumber, and the beef mixture. Fold the bottom edge over the filling, fold in the sides, and roll up tightly. Repeat with remaining ingredients.

In a small bowl, whisk together peanut butter and warm water until smooth. Add hoisin sauce, lime juice, soy sauce, sriracha (if using), and minced garlic. Whisk until well combined. Adjust consistency with more water if needed.