Heat olive oil in a large pot over medium heat. Add onion and garlic and sauté until soft and fragrant.
Stir in the flour and cook for a minute to form a light roux.
Pour in the vegetable broth and milk, whisking to smooth out any lumps.
Add broccoli and carrots, then simmer for about 15 minutes until tender.
Reduce heat to low, stir in yogurt and cheddar cheese until melted and creamy.
Season with salt and black pepper to taste. Serve warm with crusty bread if desired.