Put the cookies into the bowl of a food processor and pulse until crumbled. Add the butter and pulse until well combined. Scrape the crumbs into the bottom of a 9-inch springform pan. Using the bottom of an 8-inch cake pan, press down very hard all the way around so that the crumbs are flat and even.
Put the cream cheese and sugar into a bowl and mix together gently, this will take only a few turns of the spoon. Melt the chocolate in a large heatproof bowl set over a pan of simmering water, making sure the bottom of the bowl does not touch the water, or you can do this in the microwave. Pour a quarter of the chocolate into a small bowl, add the oil, set aside and keep warm.
Now add a large dollop of the cream cheese mixture into the large bowl of chocolate and stir well to combine. Add another dollop of cream cheese mixture to the chocolate mixture and stir to combine. Keep adding the cream cheese mixture, one dollop at a time, and mixing it like mad until the chocolate mixture begins to look uniform, smooth and silky. At this point, add the remaining cream cheese mixture into the chocolate mixture and mix together until it is completely incorporated.
Pour the mixture on top of the biscuit base and use the back of a large spoon to smooth the top. Press it down well so there are no gaps. Put the cheesecake in the refrigerator for about 20 minutes, or the freezer for 10 minutes, until the chocolate top is beginning to firm up a little. It won't be totally hard, but it will begin to feel a bit tacky.
Five minutes before the cheesecake is ready, melt the white chocolate and make sure the milk chocolate is nice and runny. You can warm it in a microwave for 20 seconds if it has hardened. Now remove the cheesecake from the fridge and, acting quickly, pour the milk chocolate over the top, spreading it out as you go until the top is completely covered. For a smooth finish, pick up the cheesecake and tilt it back and forth, letting the chocolate run over and cover the cream cheese filling. Now drizzle white chocolate lines across the top about 1/2-inch apart. The neatest way to do this is by putting the chocolate into a piping bag fitted with a very small nozzle. Place the cheesecake so the lines are running towards you, then take a toothpick and drag it from left to right, 1/2-inch apart so you have a grid. Then drag the toothpick from right to left between the lines that you have just made. You will need to do this quite quickly so the chocolate lines are still runny.
Leave the cheesecake in a cool place to set for about 2 hours.
When you are ready to serve, run a knife around the inside of the pan and remove the outside of the pan. Serve in wedges with a drizzle of heavy cream.