Step 1: In a large bowl, cream together the softened butter and granulated sugar until light and fluffy, about 2-3 minutes.
Step 2: Beat in the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
Step 3: In a separate medium bowl, whisk together the all-purpose flour, Dutch-processed cocoa powder, baking soda, and salt.
Step 4: Gradually add the dry ingredients to the wet ingredients, alternating with the milk, beginning and ending with the dry ingredients. Mix until just combined, being careful not to overmix.
Step 5: Divide the dough in half, flatten into discs, wrap tightly in plastic wrap, and chill in the refrigerator for at least 1 hour (or up to 2 days). Chilling is crucial for easy handling and to prevent spreading.
Step 6: Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper.
Step 7: On a lightly floured surface, roll out one disc of dough to about 1/4-inch thickness. Using a 2 1/2-inch to 3-inch round cookie cutter, cut out cookies. Reroll scraps and cut more cookies. Repeat with the second disc of dough.
Step 8: Place the cut-out cookies about 1 inch apart on the prepared baking sheets.
Step 9: Bake for 10-12 minutes, or until the edges are set and the centers are still slightly soft. Do not overbake, as they will become too hard. 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, prepare the ice cream. Take the softened vanilla ice cream and spread it evenly onto a baking sheet or a shallow dish lined with parchment paper, about 1/2 to 3/4-inch thick. You may need to do this in batches. Freeze the spread ice cream for at least 30 minutes, or until firm enough to cut.
Step 11: Using the same cookie cutter you used for the cookies (or slightly smaller), cut out ice cream circles. Work quickly to prevent the ice cream from melting too much.
Step 12: Place one ice cream circle on the flat side of a cookie. Top with another cookie, flat side down, gently pressing to form a sandwich. Smooth the edges of the ice cream with an offset spatula or knife if desired.
Step 13: Immediately wrap each ice cream sandwich tightly in plastic wrap or parchment paper. Place them on a baking sheet and freeze for at least 2 hours, or until very firm. Store wrapped ice cream sandwiches in an airtight freezer-safe container for up to 2 weeks.