Cut the broccoli into small florets, then trim the stalks and finely slice. Transfer to a large pan of boiling salted water and cook for 8 minutes, then add the tagliatelle and cook for a further 4 minutes. Drain and reserve the cooking liquid.
Meanwhile, heat the olive oil in a large, deep pan and gently cook the chillies, garlic and rosemary for 2 minutes. Remove from the heat, add the walnuts, broccoli and pasta and mix well.
Add the lemon juice and zest and some of the reserved pasta water and season with freshly ground black pepper. Divide among 4 plates and serve with a sprinkling of parmesan and a drizzle of olive oil.