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Trim, wash and finely slice the leeks.
Peel and finely slice the garlic and pick the thyme leaves, then place in a large casserole pan on a medium heat with the butter and 1 tablespoon of oil.
Once sizzling, stir in the leeks and 400ml of water, then cover and simmer gently over a low heat for 40 minutes, or until sweet and soft, stirring occasionally.
Season with sea salt and black pepper.
When the leeks are almost done, cook the pasta in a large pan of boiling salted water according to the packet instructions, then drain, reserving a mugful of starchy cooking water.
Toss the drained pasta into the leek pan, then remove from the heat and wait 2 minutes for the pan to cool slightly while you finely grate the cheese and beat it with the egg.
Loosen the egg mixture with a splash of reserved cooking water, then pour over the pasta, tossing vigorously.
Season to absolute perfection, going a little OTT on the pepper.
Adjust the consistency with extra cooking water, if needed, and finish with a little stroke of cheese.