Place the roughly chopped walnuts in a dry skillet over medium heat. Toast for 3–4 minutes, stirring frequently, until fragrant and lightly golden. Remove immediately and set aside.
Bring a large pot of water to a boil. Add the pasta and cook according to package instructions until al dente. In the last 3 minutes of cooking, add the broccoli florets directly to the pasta water. Reserve a generous cup of pasta cooking water before draining. Drain and set aside.
Heat the olive oil in a large skillet over medium-low heat. Add the sliced garlic and chilli flakes and cook gently for 2 minutes until soft and fragrant. Add the anchovy fillets and break them up completely with a fork — they will dissolve almost entirely into the oil. Cook for 2 minutes stirring constantly.
Add the drained pasta and broccoli directly to the skillet. Toss everything together well, adding splashes of reserved pasta water to create a silky, glossy sauce. Add the lemon zest, black pepper, and half the fresh parsley. Toss again.
Divide between two bowls. Scatter the toasted walnuts and remaining fresh parsley generously over the top. Finish with a final drizzle of extra virgin olive oil. Serve immediately.