For the double pie crust: In a food processor or quickly using your fingertips, combine the flour, sugar, salt, Cheddar and butter until the chunks of butter are broken down to the size of peas and the flour feels like wet sand. Add the first 1/4 cup of water and mix until the dough comes together easily. It's too dry if it immediately clumps apart. Add 2 more tablespoons of water at a time; you can always add more water but not more flour, so careful not to add too much!
Gather the dough into two balls and plop them on top of 2 sheets of plastic wrap. Loosely wrap up each ball and press down, smooshing the ball into a disc about an inch thick.
Pop them in the fridge for at least 30 minutes while you busy yourself with the other stuff.
For the apple filling: Preheat the oven to 375 degrees F. Combine the apples, sugars, flour, cream, lemon, vanilla, cinnamon, salt and cloves in a bowl until everything is lightly coated.
Let the dough sit at room temperature for 5 to 10 minutes. Sprinkle some flour on top of the dough as well as the surface you're rolling on. Roll out each disc to a 12-inch circle about 1/8-inch thick. As you roll out the dough, use a metal spatula to check if the dough is sticking to the surface below. Add a few sprinkles of flour if necessary to keep the dough from sticking. Roll the dough up onto the rolling pin, like a spool, and place onto a 9-inch pie dish, lining up with the center of the pan. Gently unroll and press down to line the pie dish with the dough. Fill with the apple slices, mounding a bit in the middle, and cover with the other layer of dough. Cut off the excess dough, leaving a 1-inch border around the dish. Pinch the edges to flute together (or press with a fork if that's easier) and cut three small vents into the pie.
For the egg wash: Whisk together the cream and egg in a small bowl. Brush over the top of the pie. Sprinkle with granulated sugar and bake on the center rack until golden brown, 50 to 60 minutes. (If the edges brown too quickly, cover with aluminum foil.)
Let the apple pie cool until it's just warm before cutting. Serve with vanilla ice cream.