Combine the butter, sugar, and water in a saucepan and bring everything to a boil. Add in the flour and stir the mixture with a wooden spoon until the dough becomes smooth and starts to stick to itself. Take the saucepan off of the heat and let the mixture cool down slightly, then add in the vanilla and stir in 1 egg at a time until everything is combined.
Transfer the choux pastry to a piping bag with a star-shaped nozzle and pipe it around a party hat. Transfer the party hat to the freezer and freeze it. Repeat this process with the remainder of the choux pastry.
Once the churro cones have frozen, heat a large pot filled with cooking oil on the stove and cover a plate with cinnamon sugar. Remove the party hat from inside the churro cones and deep fry them in hot cooking oil until golden brown. Place the churro cones on some paper towels and let the excess oil drain off, then roll them in cinnamon sugar.
Dice the apples and mix them together with lemon juice & zest. Bring the water, sugar, brown sugar, corn starch, apple pie spice blend, and salt to a boil in a medium-sized pot and let everything simmer for 3 minutes. Add in the apple mixture, then let everything cook together for 8 minutes. Once the apple pie filling has cooked, cover the pot and let it cool.
Heat the heavy cream and pour it over the chopped white chocolate. Stir everything together until the white chocolate has melted, then let the mixture cool down to room temperature. Whip the butter until it's creamy, then gradually stir in the white chocolate mixture until combined. Crush the graham crackers, then stir them into the mixture along with the diced apple and cinnamon. Once combined, pour the mixture into a large bowl or casserole dish, spread it around evenly, transfer it to the freezer, and freeze it.
Once the white chocolate & apple pie ice cream is frozen, spoon the apple pie filling into a churro cone, then use an ice cream scoop to scoop the white chocolate & apple pie ice cream on top. Top your apple pie churro ice cream cone off with caramel syrup to serve!