First cook the courgette. Heat the oil over a high heat in a heavy bottomed pan - you want just enough oil to coat the bottom of the pan in a shallow layer. When hot, add the courgette slices in batches (keep in a single layer and don't crowd the pan) and fry for 2-3 minutes on each side, flipping once, until just beginning to turn golden. Transfer to a plate lined with kitchen paper and sprinkle with a little coarse salt. This step can be done up to 24 hours in advance - keep cooked courgette in the fridge and allow to come to room temperature before finishing the dish.
Bring a large pot of salted water to boil and cook the spaghetti until just before al dente. Reserve a cup of cooking water.
Meanwhile, melt half the butter in a pan over medium heat and add the courgette slices. Pour over about ¼ cup of pasta cooking water and the rest of the butter and allow to cook while stirring gently for about 5 minutes.
Add the pasta to the pan with the courgette and another ¼ cup pasta water. Keep stirring everything together over a medium heat for a few minutes- as the liquid reduces it should emulsify into a glossy sauce which coats the pasta. (I prefer to use tongs to toss the pasta.)
Stir in the cheese and keep tossing until it has melted into the sauce. Add more pasta water as needed if the cheese clumps together.
Season with black pepper and extra cheese to serve.