Bring a large pot of water to a rolling boil. Season liberally with salt, it should taste like the sea.
Add butter to a large skillet. Turn the heat on medium and once butter melts, add onion, carrot and 1/2 teaspoon salt. Sweat the onions and carrots until softened, about 4-5 minutes. Whisk in flour. Cook for 1 minute.
Mix chicken stock, heavy cream and milk in a large measuring cup. Slowly whisk mixture into butter, flour and vegetable mixture making sure there are no lumps. I like to add about 1/3, and whisk it in completely before adding the rest, which helps to prevent lumps. Add another teaspoon of salt, nutmeg and pepper, whisk. Bring to a boil and reduce to a simmer until thickened, about 5-6 minutes. Once the sauce is thickened, remove from the heat and working in batches, stir the cheese until melted. I like to add the cheese in thirds, stirring until each batch has melted before adding the next. This helps to ensure a creamy sauce.
While the sauce thickens, add pasta to the boiling water. Once it cooks for a couple of minutes, add the broccoli. When the broccoli is soft and bright green and the pasta is aldente, drain.
Place pasta and broccoli back in the pot and then pour the cheese sauce along with the remaining salt on top. Toss until combined. Season with salt, pepper and nutmeg. Serve with crusty bread.