Cut steak diagonally across the grain into 16 (1/8- to 1/4-inch-thick) slices.
Stir together 2 tablespoons sugar and next 5 ingredients in a large bowl. Add meat, stirring to coat. Cover and chill 30 minutes. Remove meat from marinade, discarding marinade. Sprinkle steak evenly with salt and pepper.
Grill flank steak slices over medium-high heat (350° to 400°), uncovered, 2 minutes on each side or until done. Cover with aluminum foil to keep warm.
Stir together 1 1/4 cups hot water, fish sauce, 1/3 cup sugar, and vinegar until sugar dissolves. Stir in 1/3 cup carrot. Set aside.
Pour hot water to a depth of 1 inch in a large shallow dish. Dip 1 spring roll wrapper in hot water briefly to soften; pat dry with paper towels.
Combine remaining 2/3 cup carrot, bean sprouts, and next 4 ingredients. Place 1 slice of beef on one side of wrapper; top with 1/3 cup lettuce. Place about 1/3 cup bean sprout mixture on lettuce on wrapper. Fold sides of wrapper over filling, and roll up, jelly-roll style. Serve with fish sauce mixture for dipping.
Note: Prepare fresh lemon grass as you would a green onion--peel off the outer layer, remove the root end, and use the white portion. Fish sauce and rice paper wrappers can be found on the ethnic foods aisle of most grocery stores.