For the ice cream: Melt the butter in a large skillet over medium heat; cook, stirring occasionally, until the milk solids turn a pecan-brown color, about 5 minutes. Stir in the apples, cinnamon and a pinch of salt, and continue to cook, stirring occasionally, until the apples have completely softened, 10 to 15 minutes. Add the granulated sugar, lemon juice and 1 tablespoon water. Simmer until the sugar has dissolved and the liquid is syrupy, about 3 minutes. Let cool to room temperature, about 20 minutes.
While the apple mixture cools, take the ice cream out of the freezer and let soften. Quickly stir the cooled apple mixture into the softened ice cream, and freeze until hardened, about 30 minutes.
For the cookies: Meanwhile, position 2 oven racks in the center of the oven, and preheat to 325 degrees F. Line 2 baking sheets with parchment.
Whisk together the flour, baking soda, 1/4 teaspoon salt and cinnamon in a medium bowl.
Whisk together the melted butter, brown sugar and corn syrup in a large bowl until smooth. Add the egg and vanilla, and whisk until combined. Add the flour mixture, and stir with a rubber spatula until just incorporated. The dough will be quite loose; refrigerate it for 15 minutes to let it firm up a little.
Scoop out 12 mounds of dough, about 2 scant tablespoons each, and place them at least 2 inches apart on the prepared baking sheets. (The cookies will spread a lot during baking.) Lightly flour your hands, and press the dough to about 1/4 inch thick. Bake, switching the sheets halfway through, until the cookies are lightly browned at the edges, 10 to 12 minutes. Let the cookies rest on the sheets for 5 minutes, then carefully transfer to a rack to cool completely, about 20 minutes.
To assemble: Turn 6 cookies upside down on a cutting board. Scoop about 1/3 cup of slightly softened apple pie ice cream onto each cookie, and spread with a spoon or a butter knife to the edge. Top with a second cookie, and press down gently to adhere. Serve the ice cream sandwiches right away, or put them in the freezer to harden for about 1 hour, then wrap and freeze for up to 2 weeks.
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