Food colourings: red, orange, yellow, green, blue, dark purple, pink
Grease the cake tin and line the bottom. Wrap a few layers of foil around it - to make it watertight - and place it in a larger oven dish or tray so that it’s sitting in a water bath. Don’t add water yet. Boil the kettle and preheat the oven to 200C (180C fan).
Put the cream cheese, sugar and butter into a mixing bowl and whisk together. Add the egg yolks and continue beating, then add the flours and salt, then milk, lemon zest and juice. The mixture should be smooth with no lumps. Divide the mixture between 7 bowls and colour each one a different colour of the rainbow: Pink - 25 grams mixture; Dark purple - 50 grams; Blue - 75 grams; Green - 100 grams; Yellow - 125 grams; Orange - 150 grams; Red - 175 grams.
Add enough colouring to make the colours really vibrant (they’ll be paler once the egg whites are mixed in) then put to one side while you whisk the egg whites, cream of tartar and remaining caster sugar to stiff peaks. Add egg whites to each colour mixture in the following quantities. Mix only as much as you need to and be careful not to knock too much air out: Pink - 10 grams; Dark purple - 20 grams; Blue - 30 grams; Green - 40 grams; Yellow - 50 grams; Orange - 60 grams; Red - 70 grams.
Start layering your cake tin: spoon all of the red cheesecake filling into the cake tin. Then gently spoon the orange filling in the middle of that, followed by yellow, then the rest of the colours - with pink being the final colour. Gently wiggle the tin as you go to help the mixture lay flat. If you want to create a further pattern on top, use a skewer to draw lines from the middle to the outside of the cheesecake.
Cover the top of the cheesecake tin with foil, leaving enough space for it to rise a lot. Carefully slide it, in its water tray, onto a shelf in your preheated oven. Use water from the kettle to fill the water bath - to around two-thirds up the cheesecake. Close the door then bake for 20 minutes at 200C (180C fan), before turning the heat down to 140 C (120C fan) and cooking for another 1 hour. Then turn the oven off and leave the door closed for 30 minutes.
Bring the cheesecake out of the oven and run a knife around the edge to loosen, if needed. Get two plates and lightly oil one so that the top of the cake doesn’t stick to it. Turn the cake over onto this plate and remove the tin, then gently turn it right-way up onto a serving plate. Serve immediately.