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Boil the kettle. Heat ½ tbsp oil in a large, deep frying pan or saucepan over a low-medium heat and sizzle the pancetta until crisp, about 5 mins. Once crisp, reduce the heat to as low as it will go to keep warm.
Meanwhile, put the parsley, basil, mint and remaining oil in a small food processor and blitz until finely chopped. Or, finely chop the herbs using a sharp knife and omit the oil. Whisk the egg yolks and parmesan together in a separate bowl.
Fill a large saucepan with boiling water from the kettle, season well with salt and bring to the boil on the hob, then cook the spaghetti for 10 mins until al dente. Add the asparagus and broccoli for the final 3 mins of cooking time, and the peas for the final 30 seconds.
Lift the spaghetti into the pan with the pancetta using tongs, allowing a little water to be added with each addition of pasta. Drain the boiled veg but reserve a mug of the cooking water, then add the veg to the pan with the spaghetti and pancetta. Keep the heat set as low as possible and ensure there is a small puddle of the pasta cooking water in the bottom of the pan. This will prevent anything catching and help loosen the sauce.
Scrape all of the herb mixture and the egg and cheese over the warm spaghetti mixture and use tongs to toss everything together, ensuring each strand of spaghetti is well-coated. Continue tossing the pasta through the sauce for a few minutes, turning the heat up slightly if needed until the sauce thickens and clings to the spaghetti. Add splashes of the reserved pasta cooking water if the sauce becomes too thick – the spaghetti should be glossy and shiny when ready.
Divide between bowls and top with extra parmesan, a grinding of black pepper and a drizzle of extra virgin olive oil, if you like.
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