1. In a large pot or Dutch oven, melt butter over medium heat. Add the chopped onion and cook until translucent, about 4-5 minutes. Stir in the minced garlic and cook for 30 seconds until fragrant.

2. Sprinkle flour over the onion-garlic mixture. Whisk constantly for 1-2 minutes to form a smooth roux.

3. Slowly pour in broth while whisking, then stir in half and half. Blend until smooth.

4. Add chopped broccoli and shredded carrots. Simmer for 15-20 minutes uncovered, until broccoli is tender and soup thickens slightly.

5. For a smoother texture, use an immersion blender or blend half of the soup and return it to the pot. This step is optional.

6. Reduce heat to low and stir in freshly grated cheddar until melted. Season with salt and pepper to taste. Serve hot, garnished with extra cheese or broccoli florets.