Mediterranean Pasta with Broccoli

1 1/2 pounds broccoli, stalks peeled and sliced crosswise, florets broken into small pieces
3/4 pound elbow macaroni
1/4 cup extra-virgin olive oil
1/2 cup coarsely chopped walnuts
3 large garlic cloves, minced
4 oil-packed anchovy fillets, drained
1/4 cup finely grated Parmigiano-Reggiano cheese
Salt and freshly ground pepper

Bring a large pot of salted water to a boil. Add the broccoli stalks and cook for 3 minutes, then add the pasta and cook, stirring occasionally, for 5 minutes. Add the broccoli florets and continue cooking until the pasta is al dente. Drain the pasta and broccoli and return them to the pot.
Meanwhile, heat the olive oil in a large, heavy skillet. Add the walnuts and cook over moderate heat, stirring, for 2 minutes. Add the garlic and anchovies and cook until the anchovies break down, about 2 minutes longer. Add the drained pasta and broccoli and toss well. Sprinkle with the Parmigiano, season with salt and pepper and serve.