Cook the pasta in a large pot of generously salted boiling water according to the packet instructions. Reserve a cup of the cooking water before draining.
While the pasta cooks, blanch the broccoli florets in a separate pot of salted boiling water for about 3 to 4 minutes until just tender but still firm. Drain and set aside.
Heat the olive oil in a wide pan over medium heat. Add the anchovies and allow them to melt and dissolve into the oil, stirring gently. Remove the pan from the heat, then add the crushed garlic and chilli and toss for a few seconds. Do not let the garlic brown.
Add the broccoli and toss well until the florets are coated in the flavoured oil.
Add the cooked pasta and a splash of the reserved cooking water. Toss everything together until well combined.
Serve immediately with freshly grated Parmesan.