1. Snap off tough ends of asparagus. Cut into 2-inch pieces. Cook asparagus in boiling water to cover 3 minutes. Drain, and plunge asparagus into ice water; drain. Place vinegar and next 5 ingredients (through garlic) in a large heavy-duty zip-top plastic bag. Seal bag and shake to blend. Add asparagus to bag; seal bag and shake to coat. Chill 3 hours.
2. Add carrot to bag; seal bag and shake to coat. Chill 1 hour.