Line 2 baking sheets with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter, dark brown sugar, and granulated sugar at medium speed until light and fluffy, about 3 to 4 minutes, stopping to scrape sides of the bowl. Beat in egg and vanilla until combined.

In a medium bowl, whisk together flour, kosher salt, apple pie spice, ground cinnamon, baking soda, and cream of tartar. With the mixer on low speed, gradually add flour mixture to butter mixture, beating until just combined and stopping to scrape sides of the bowl.

Using a 1½-tablespoon spring-loaded scoop, scoop dough and roll into balls. Roll dough balls in turbinado sugar, and place 1½ to 2 inches apart on prepared pans. Refrigerate for at least 30 minutes or up to 1 hour.

Preheat oven to 350°F (180°C).

Bake until edges are set and center is still slightly soft, 13 to 16 minutes. Let cool on pans for 20 minutes. Remove from pans and let cool completely on wire racks.

Place Apple Butter Buttercream in a pastry bag fitted with a ½-inch round piping tip. Pipe a buttercream border onto flat side of half of the cookies. Spoon apple butter in center. Top with remaining cookies, flat side down. Refrigerate in an airtight container for up to 3 days.