As the onions and mushrooms are cooking and the pasta water is coming to a boil, we place the butter into a pot and begin melting it over low heat.
The pasta water has come to a boil so we add some salt and the pasta. We stir and time the pasta for a couple of minutes shy of the package directions. We will keep checking the pasta to see when it is al dente.
As soon as the pasta goes in the pot, we see that the mushrooms have released their liquid and are browning, so we add the rinsed broccoli and garlic. We stir and add the dried oregano and marjoram, crushing them in our hands as we add them to the pan. We add some salt and pepper and the wine and reduce the heat to medium-low. We partially cover and cook for 5 minutes.
The butter has melted in our pot and we add the flour. We stir with our “roux whisk” (a flat whisk – but a wooden spoon and regular whisk would work) and cook over medium-low heat for 2 minutes. Meanwhile, we place the milk in the microwave for 1 minute to heat. When the flour-butter mixture has cooked for 2 minutes, we add the milk and whisk to remove all lumps. We raise the heat slightly and bring the mixture to a boil. We lower the heat and simmer gently for a couple of minutes.
When the pasta is cooked, we drain it in a colander. We pour the thickened milk mixture into the simmering vegetables and add most of the cooked pasta (as much as will fit in the pan). We add the cherry tomatoes and grate some parmesan into the pan with our microplane grater. We taste for salt and pepper and reseason. We serve topped with strips of the prosciutto and more parmesan cheese.