Make the pastry in a food processor as normal: pulse the butter and shortening into flour until it looks like crumbs. Leaving the mixture in the bowl, put the grating blade in and process the cheese into the crumb mixture. Replace the normal blade and add the egg, ice water and salt to bind. Turn the pastry out and press it into two discs, one slightly smaller than the other. Wrap each in plastic wrap and put them in the refridgerator to rest for at least 20 minute
Preheat the oven to 400* and put in a baking sheet. Slice the McIntoshes into small chunks and fry in half the butter until they become soft and begin to lose their shape. Add the cloves and nutmeg. Tip the apple mush into the food processor and puree, pulsing so as not to make it too like baby food. Add about 3/4 of the beaten egg and all of the sugar and pulse again to mix. Fry the Cortlands in the other half of the butter and cover them to help them cook a little. Cook for about 10 min: they should be tender but still holding their shape.
Roll out the larger disc of pastry between two plastic wrap and line an 8" springform pan with it, letting it hang over the sides. Pour in the pureed mixture and then push the Cortland pieces into the puree to coat them. Roll out the smaller disc of pastry between plastic wrap to form the top. Lay over the pie and curl the edges inward, crimping them to seal. Decorate with any leftover pastry if you wish, and brush with the remaining egg as a glaze. Cook for 15 min, then turn the oven down to 350* for a further 30 minutes. Sprinkle with sugar when it comes out of the oven.