Separate the stalks and the florets. Split in half the stalk and set aside.
Bring a large pot of salted water to a boil and add the pasta. Boil for 8 minutes or until very almost cooked through, then add the broccoli florets. Cook until the broccoli is tender, and the pasta is cooked. Reserve 1 cup of the boiling water and drain the pasta.
Meanwhile, in a large pan, melt butter and add the garlic and broccoli stems and cook for a few minutes over medium-high heat. . Pour the cream, lemon juice, and zest. Season to taste. Allow simmering for a few minutes until the sauce has thickened slightly.
Add the drained pasta and broccoli into the sauce and add a splash of the cooking water. Toss to coat in sauce, adding more pasta water if necessary.
Add a generous amount of grated cheese, toss, and serve.