1. Melt the butter in a large heavy bottomed pot over medium heat. Add the chopped onion and shredded carrots with a pinch of salt and cook, stirring occasionally, until the vegetables are softened and the onion is translucent, about 5 to 7 minutes.

2. Stir in the minced garlic and cook for 30 seconds, just until fragrant, being careful not to let it brown.

3. Sprinkle the flour evenly over the vegetables and stir well so everything is coated in the buttery mixture. Cook this roux for 1 to 2 minutes, stirring constantly, to remove the raw flour taste.

4. Slowly pour in the chicken or vegetable broth while whisking continuously to avoid lumps. Once smooth, whisk in the half and half and bring the mixture to a gentle simmer over medium low heat, stirring often.

5. Add the chopped broccoli, Dijon or dry mustard, nutmeg, salt, black pepper, and cayenne or hot sauce if using. Reduce the heat to low and let the soup simmer gently for 15 to 20 minutes, stirring occasionally, until the broccoli is tender and the soup has thickened.

6. For a smoother texture, use an immersion blender to puree part of the soup directly in the pot, leaving some small broccoli pieces for texture. You can skip this step if you prefer a chunky soup.

7. Turn the heat to the lowest setting and add the shredded cheddar cheese in small handfuls, stirring after each addition until it is fully melted and the soup is creamy. Do not let the soup boil once the cheese is added.

8. Taste and adjust the seasoning with additional salt, pepper, or hot sauce as needed. If the soup is too thick, thin it with a splash of extra broth or half and half until it reaches your desired consistency.

9. Ladle the hot soup into bowls, garnish with a little extra shredded cheddar, and serve immediately with crusty bread or bread bowls for dipping.