In a large pot over medium heat, melt the butter. Add the chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and cook until fragrant.

Add diced carrots and broccoli florets. Sauté for another 5 minutes until slightly tender.

Pour in the vegetable broth and bring to a boil. Reduce heat to low and simmer for about 10 minutes until vegetables are fully cooked.

Use an immersion blender to blend the soup until smooth but slightly chunky.

Return to low heat, stir in heavy cream, and gradually add grated cheddar cheese while stirring until melted.

Season with salt and pepper to taste. Serve hot with crusty bread or crackers.