In a large pot or Dutch oven, melt the butter over medium heat. Add the chopped onion and sauté until it becomes translucent, about 5 minutes. Add the minced garlic and cook for another minute until fragrant.

Stir in the flour and cook for about 2-3 minutes, stirring constantly to form a roux. The roux should be lightly browned but not burnt.

Gradually whisk in the half-and-half and chicken broth. Continue whisking until smooth and well combined.

Add the chopped broccoli and grated carrot to the pot. Bring the mixture to a simmer and cook for about 15-20 minutes, or until the vegetables are tender.

For a smoother texture, use an immersion blender to blend the soup to your desired consistency. You can also transfer the soup to a blender in batches, being careful with the hot liquid. If you prefer a chunkier soup, you can skip this step or blend only part of the soup.

Return the soup to low heat. Stir in the shredded cheddar cheese until it melts and is fully incorporated. Season with salt, pepper, and nutmeg (if using) to taste.

Ladle the soup into bowls. Garnish with additional shredded cheddar cheese and serve with crusty bread or in bread bowls for a classic Panera-style experience.