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Cook the spaghetti in a large pan of boiling salted water, according to packet instructions. Cut the broccoli into small florets, thinly slice the thick stalks and throw into the pan of pasta for the last 3 minutes of cooking time.
Meanwhile, heat 3 tablespoons of olive oil in a frying pan, add the anchovies and chillies and fry briefly. Add the breadcrumbs and cook, stirring, for about 5 minutes until the crumbs are crunchy and golden.
Drain the spaghetti and return to the pan. Toss with three quarters of the crumb mixture, some salt and pepper and another 2 tablespoons of olive oil. Serve each portion sprinkled with the remaining crumbs.