In the work bowl of a food processor, combine the flour and salt. Pulse together 5 times.
Add in the chilled butter and lard, then pulse quickly 8 times.
Mix the vodka and water together in a measuring cup, then slowly pulse it into the work bowl.
The pie dough should slightly come together but still be shaggy.
Gather it up and form 2 equal disks.
Wrap and chill for half an hour.
Roll out the first disk and place it in a 10-inch cast iron or deep pie dish. Place it back in the refrigerator to chill while making the filling.
In a small saucepan, combine the bourbon and 1 cup of the sliced apples. Cook over medium heat, stirring occasionally, until the apples are soft and the bourbon has dissipated.
In a separate bowl, mix together the flour, cinnamon, salt, and nutmeg.
Toss the remaining raw apples with lemon juice, then mix in the dry ingredients.
Stir in the bourbon apples and place the mixture in the prepared pie dish.
Roll out the top crust and place it over the pie.
Seal the edges, create a vent, and crimp the pie.
Mix the remaining sugar and cinnamon together. Brush the half and half lightly all over the top crust. Sprinkle cinnamon sugar evenly over the top of the pie. Place the pie in the refrigerator while preheating the smoker.
Preheat the smoker to 425°F (220°C).
Once at temperature, place the pie inside and close the lid. Smoke for 20 minutes.
Reduce the heat to 375°F (190°C) and continue smoking for 30 minutes. Lightly tent the pie with foil and bake for an additional 22-25 minutes.
Check for doneness. The pie should be golden and bubbling. It may need an additional 8-10 minutes longer. Let the pie cool slightly before slicing.
In a saucepan, combine the heavy cream and butter over medium heat. Once the butter melts, whisk in the remaining ingredients.
Bring the mixture to a boil, then reduce the heat to a simmer.
Let it simmer while whisking for 10-12 minutes until it thickens and darkens slightly.
Remove from heat and mix in the vanilla bean caviar and extract.
Pour the sauce into a large 4-cup measuring cup. The sauce will thicken as it cools.
Use the miso caramel sauce as a drizzle and serve with ice cream.