Preheat the oven to 400°F.
To make the sour cherry filling, heat a small saucepan over medium heat with sour cherries, granulated sugar, lemon zest and lemon juice. Once the mixture starts to boil, turn heat down and simmer for about 8 to 10 minutes. Once slightly reduced, combine cornstarch with water into a slurry and whisk it into the sour cherry mixture and let the sauce thicken up, about another 1 to 2 minutes. Once done, the mixture should be glossy and thick. Set this aside to cool at room temperature.
To make the apple filling, add apples to a medium saucepan over medium heat. Add lemon juice, brown sugar, cinnamon, nutmeg and sea salt. Once the mixture starts to boil, turn the heat down and simmer for about 8 to 10 minutes. Once slightly reduced, combine cornstarch with water into a slurry and whisk it into the apple mixture and let the sauce thicken up, about another 1 to 2 minutes. Once done, the mixture should be thicker and the apples cooked down. Set this aside to cool at room temperature.
To make the pie cookies, lightly flour your work surface and unroll one of the sheets of puff pastry or pie crust. Take one sheet and mark out every ½-inch across one side. You will use a pizza slicer to cut out strips to make your lattice pattern. Once cut, set these aside over the edge of a bowl.
Lay out the second sheet of pastry and add the sour cherry filling in the center. Using an offset spatula, spread out the filling evenly on the pastry, leaving ¼-inch bare from the edges.
Using the strips, make a lattice pattern by layering the strips on top of the filling, about a ½-inch apart, in an over under pattern. Trim any of the overhang if needed with a dough cutter. Fold the bare edges over to create a border around the lattice.
Use a 3-inch round cookie cutter to cut out 8 or 9 cookies. Transfer the cookies to a parchment lined baking sheet. Brush the lattice with nondairy milk. Sprinkle demerara sugar on top of each cookie. Bake for 14 to 16 minutes until the dough is puffy and cooked through. Transfer to a cooling rack and let them cool completely.
To make the ice cream filling, remove the top of the ice cream pint. Use sharp kitchen scissors to cut into the side of the container. Remove the container gently to keep the ice cream intact. Turn the ice cream so it’s sitting on its side. Use a large knife to slice the round of ice cream into ½-inch thick discs.
Since the ice cream container gets slightly smaller towards the bottom, the ice cream filling will vary slightly in size. To get the sizes perfectly uniform, use the 3-inch round cookie cutter to punch out the ice cream rounds before placing on the pie cookie.
Place each ice cream round on a pie cookie. Then top with another pie cookie to make pie cookie ice cream sandwiches.
Eat immediately or freeze the assembled ice cream sandwiches. You can eat them pretty much out of the freezer, as the crust stays soft and delish!