Add the olive oil and lardons to a frying pan set over a medium heat. Fry for 5 minutes until the lardons begin to crisp then add the mushrooms and fry for 2 minutes more. Stir in the garlic for a last minute of cooking. Keep warm.
Whisk the egg yolks, egg and two-thirds of the grated cheese together in a small bowl. Season with plenty of black pepper.
Cook the pasta in plenty of salted water, according to pack instructions. Save a quarter of a mug of pasta cooking water before draining it. Return the drained pasta and mushroom mixture to the saucepan and add the egg and cheese mixture, stirring constantly. Add enough of the reserved pasta water to create a creamy sauce that coats the pasta, about 4 tablespoons. Stir in the basil. Serve with the remaining grated cheese and a few extra basil leaves.
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If serving vulnerable groups, elderly people, babies, toddlers, pregnant women and people who are unwell, cook until the whites and yolks are solid.