Heat the oil and 15g butter in a large frying pan or sauté pan over a medium-high heat. Add the squash, cover with a lid and cook, stirring regularly, for 10 minutes until the squash is tender and turning golden.
Meanwhile, cook the pasta in a large pan of salted water for 1 minute less than the pack instructions. Before draining, scoop out a mugful of the cooking water and set aside.
Reduce the heat under the squash to medium and remove the lid. Add the shallots and garlic; season. Fry, stirring often, for 5 minutes until soft. In a separate pan, cook the remaining 45g butter over a high heat, swirling occasionally, until nutty and golden. Add the sage and remove from the heat.
Tip the cooked pasta into the pan with the squash and add the brown butter, lemon juice and a glug of the cooking water. Stir in the cheese and toss to coat over the heat for 1 minute, adding more cooking water to loosen the sauce if needed (it should be glossy and just coat the pasta). Divide between plates and serve with extra cheese on top.