Fill a large pot with water and heat it over high heat until boiling. Once the water boils, add the kosher salt and the noodles. Cook only until al dente, 9-10 minutes or whatever the package directions tell you.
While the pasta cooks, gently melt the butter over medium heat in a large, non-stick skillet. Once the butter is melted, add in the garlic and cook until fragrant.
Drain the pasta (reserving the water) and transfer the pasta to the skillet with the melted butter.
Reduce the heat to medium low. Add a handful of grated cheese and a ladle of hot pasta water, tossing the pasta continuously with your tongs to incorporate the cheese. At this point, you will add in the warm whole milk and continue to toss together.
Add more cheese, a little at a time, tossing between each addition, until all the cheese has been used. Add more hot pasta water as needed to melt the cheese so that it coats the noodles but is not gloppy. (But be careful not to add too much. You don’t want a watery sauce.)
Add in the broccoli and toss to coat with the pasta and the sauce. Taste the pasta and adjust the seasoning as needed. You will want to serve this immediately. As it cools it will clump up on you. So, eat it hot and freshly made. ENJOY!