1. Melt the butter in a large pot over medium heat, then add the diced onion and cook until soft and fragrant.

2. Stir in the garlic and cook briefly until aromatic.

3. Sprinkle in the flour and stir constantly until it forms a smooth paste with the butter and onions.

4. Slowly whisk in the chicken broth until the mixture is smooth.

5. Add the half-and-half and continue whisking until the soup base begins to thicken.

6. Stir in the broccoli, carrots, ground mustard, paprika, nutmeg, salt, and black pepper.

7. Simmer gently until the broccoli is tender, stirring occasionally.

8. Blend part of the soup with an immersion blender if you want a creamier texture while leaving some broccoli pieces whole.

9. Reduce the heat to low, then add the cheddar cheese a handful at a time, stirring until melted.

10. Stir in the Parmesan cheese, taste, adjust seasoning, and serve warm with crusty bread or a bread bowl.