1. Heat a large saucepan over medium-high heat. Add oil to pan; swirl to coat. Add onion, carrot, and tomato paste. Cook 10 minutes, stirring occasionally. Add garlic; cook, stirring constantly, 1 minute. Stir in flour; cook 1 minute, stirring frequently.
2. Stir in 1 1/4 cups water, half-and-half, and clam juice. Bring to a boil; reduce heat, and simmer, uncovered, 10 minutes, stirring occasionally. Remove from heat; add cheese, stirring until melted. Stir in sherry, chives, and thyme. Ladle soup into bowls; top evenly with crab. Garnish with chopped chives, if desired.