Cook the pasta in salted water according to package instructions until al dente. Drain and set aside.

Chop the onion, carrot, celery, and garlic finely or use a food processor.

In a large pot, melt butter over low heat. Add chopped vegetables and sauté until soft, about 5 minutes.

Stir in flour and black pepper, stirring until smooth.

Gradually add chicken broth, milk, nutmeg, and dry mustard, whisking continuously.

Increase heat to medium-low, bring to a gentle boil, then cover and cook for 10-15 minutes.

Add broccoli florets and Parmesan cheese, mixing well. Season with salt and pepper to taste.

Cook uncovered until broccoli is tender, about 5 minutes. Remove from heat.

Gradually add grated cheddar cheese, stirring until melted into the soup.

Return the cooked pasta to the soup, stir well, and adjust seasoning if needed.

Serve immediately to prevent pasta from absorbing too much broth.