Prep: 5 minutes; Cook: 20 minutes; Total time: 25 minutes.
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.
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.
Preheat broiler to medium-high heat. Place rack 4 inches from heat.
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.
Add more stock to soup if too thick; cook until hot. Pour over bread; sprinkle with Parmesan. Top with thyme, if desired.
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