Sauté Vegetables: In a large pot over medium heat, melt the butter and add the chopped onion. Cook until translucent, about 5 minutes, then stir in the minced garlic and let it cook until fragrant.

Combine Carrots and Broccoli: Add the diced carrots and fresh broccoli florets to the pot. Sauté for another 5 minutes until they start to become slightly tender.

Add Broth: Pour in the vegetable or chicken broth and bring everything to a boil. Reduce the heat and let it simmer for about 10 minutes, allowing the vegetables to cook through.

Blend Soup: Using an immersion blender, blend the soup until it's smooth yet retains a slightly chunky texture.

Incorporate Cream and Cheese: Return the blended soup to the pot and stir in the heavy cream. Gradually add the grated sharp cheddar cheese, mixing until it's completely melted and creamy.

Season and Serve: Add salt and pepper to taste, then ladle the soup into bowls. Serve hot with crusty sourdough bread or alongside your favorite crackers.