Step 1Bring a large pot of generously salted water to a boil.
Step 2Trim the broccoli: Remove the florets and peel and dice the stems, keeping them separate. You should have about 5 cups total.
Step 3Chop coarsely together by hand the garlic, anchovy fillets, capers, olives and pistachios.
Step 4Heat the oil gently in a skillet large enough to hold the pasta later. Add the chile and discard when it begins to color. Add the garlic mixture to the pan and cook gently in the oil until it just begins to turn gold, about 2 minutes.
Step 5When the water is boiling rapidly, add the broccoli stems and cook for 2 minutes. Add the florets and continue cooking until they are bright green and tender, but still slightly crisp and not mushy, 4 to 5 minutes.
Step 6Lift the cooked broccoli out of the pot with a slotted spoon or spider strainer right into the skillet, leaving the water boiling in the pot. Stir the broccoli and garlic mixture together, breaking up any large florets with the spoon; the broccoli pieces should be small enough to coat the pasta. Taste the broccoli mixture and add more salt if necessary (with anchovies, olives and capers, you will probably not need any), and let the flavors blend for a couple minutes over low heat.
Step 7Meanwhile, add the pasta to the boiling water and cook, stirring occasionally, until it is al dente, generally 8 to 10 minutes.
Step 8When the pasta is done, lift it out of the water and transfer it, rather wet, to the skillet. Mix well over low heat for about 30 seconds, sprinkle with the cheese and mix again. Transfer to a warm serving dish or serve directly from the skillet. Serve immediately.