Grate the courgettes coarsely , then place, a fistful at a time, into a brand-new J-cloth or a clean tea towel. Squeeze out the excess water.
Put everything except the Parmesan and the pasta into a food processor and whizz until you have a green paste. Stop a couple of times to push down the courgettes and basil from the sides of the bowl. Season. Scrape the paste into a dish and stir in the cheese. Taste for seasoning.
Cook the pasta in boiling salted water until al dente. Drain, reserving some of the cooking water, and stir the pasta and pesto together. Add as much of the pasta cooking water as you need to let down the pesto a little so that it’s softer and looser. Serve immediately.