Step 1: In a medium bowl, whisk together the all-purpose flour, Dutch-processed cocoa powder, baking soda, cream of tartar, and salt. Set aside.
Step 2: In a large bowl, using an electric mixer, cream together the softened unsalted butter and granulated sugar until light and fluffy, about 2-3 minutes.
Step 3: Beat in the large egg and pure vanilla extract until well combined, scraping down the sides of the bowl as needed.
Step 4: Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix; the dough will be thick.
Step 5: Divide the dough in half. Roll each half between two sheets of parchment paper to about 1/4-inch thickness. Chill the flattened dough in the refrigerator for at least 30 minutes, or up to 2 hours.
Step 6: Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone mats.
Step 7: Remove one sheet of chilled dough from the refrigerator. Using a 2-inch square or round cookie cutter, cut out shapes. Carefully transfer the cookies to the prepared baking sheets, leaving about 1 inch between them. Repeat with the second sheet of dough. Re-roll any scraps if desired.
Step 8: Bake for 8-10 minutes, or until the edges are set and the centers look just slightly soft. Do not overbake.
Step 9: Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.
Step 10: Once the cookies are completely cool, retrieve your slightly softened ice cream from the freezer. It should be soft enough to scoop but not melted.
Step 11: Take one cooled cookie, scoop a generous amount (about 1/4 to 1/3 cup) of ice cream onto its flat side.
Step 12: Place another cookie on top, flat side down, and gently press to spread the ice cream to the edges.
Step 13: Immediately wrap each assembled ice cream sandwich tightly in plastic wrap.
Step 14: Place the wrapped ice cream sandwiches in an airtight freezer-safe container. Freeze for at least 2-4 hours, or preferably overnight, until firm.