1. Heat 2 tablespoons oil in a large, deep pot over medium heat. Add tomato paste; cook 1 minute. Add onion, carrot, salt, and pepper; cook until onion softens (3-5 minutes), stirring often.
2. Add tomato and thyme; cook until tomato breaks up (10-15 minutes), stirring occasionally. Add broth and sugar; stir. Cook until hot; simmer 5 minutes.
3. Preheat broiler to medium-high heat. Place rack 4 inches from heat.
4. Brush both sides of bread lightly with remaining oil; broil until golden (5 minutes). Rub sides of bread with garlic; break into pieces. Divide among bowls.
5. Add more stock to soup if too thick; cook until hot. Pour over bread; sprinkle with Parmesan. Top with thyme, if desired.