Toss together the lemon juice, apple pie spice, apples and 1/3 cup of the sugar in a medium bowl.
Melt 2 tablespoons of the butter over high heat in a medium skillet. Add the apple mixture and stir to coat with the butter. Lower the heat to medium, cover and cook, stirring occasionally, until the apples are tender and translucent, about 20 minutes; if the mixture becomes too dry and begins to brown before that, add 1/4 cup water and continue to cook.
Mix the cornstarch with 1 teaspoon water in a small bowl and pour into the apple mixture. Bring the mixture to a boil to activate the starch and cook until the thick and bubbly, about 1 minute more. Transfer to a bowl and reserve.
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper.
Whisk the flour, baking powder and salt together in a bowl. In another bowl, whip the remaining 12 tablespoons butter and 2/3 cup sugar with a hand-held mixer until fluffy, about 5 minutes. Beat in the egg until just combined, then mix in the vanilla. Slowly beat in the dry ingredients in 2 additions, mixing just until incorporated.
Scoop 1 tablespoon of the dough for each cookie and roll into a ball. Place about 2 inches apart on the prepared baking sheets and press down into flat discs about 1/4 inch thick. Bake the cookies until just set on top but still pale, about 15 minutes. Cool the cookies on the baking sheets.
When the cookies are cool, flip them over. On half of the cookies, spread each with 1 teaspoon apple filling. On the other half of the cookies, spread each with 1 tablespoon vanilla ice cream. Sandwich each apple filling-topped cookie with an ice cream-topped cookie and press together gently.
Serve immediately or store in the freezer for up to 3 months. Let soften 5 minutes before serving from frozen.
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