In the bowl of a blender or food processor, puree all the ice cream ingredients until smooth. Transfer the mixture to a medium bowl and press plastic wrap against the surface. Place in the refrigerator for at least 1 hour, or until chilled. Mixture can be stored for up to 1 day.

Pour the chilled mixture into an ice cream maker and freeze according to the manufacturer’s directions. Remove to an airtight container and freeze for at least 3 hours, or until firmed.

In a large bowl, use an electric mixer to beat the butter and sugar on medium-high speed until smooth and well combined, 1 to 2 minutes. Add the eggs, one at a time, until well combined. Add the vanilla extract and 1 tablespoon milk. Add the flour, baking powder, and salt.

Remove half the dough and place in a separate bowl. Each half of dough should weigh about 430 grams. Add the cocoa powder and remaining 2 teaspoons milk to half the dough and mix until combined.

Place one half of the soft dough in between two large pieces of parchment paper or plastic wrap on a work surface. Roll the dough out to a 1/4-inch thickness, using the parchment paper to prevent sticking. Repeat with the remaining half. Place each sheet of dough in the freezer until firm, 30 minutes to 1 hour.

Line large baking sheets with parchment paper or silicone baking mats. Use a cookie cutter to cut out circles from the dough and place on prepared baking sheets, spreading at least a 1/2 inch apart.

Chill the baking sheets in the refrigerator for 30 minutes, or until the dough circles are firm.

Meanwhile, preheat the oven to 350°F.

Remove the baking sheets from the refrigerator and prick the cookies all over with the probe of an instant-read thermometer or the blunt end of a skewer or chopstick. Bake the vanilla cookies first, for 9 to 10 minutes, or until the cookies are set. Then bake the chocolate cookies. Let cool on baking sheets for 5 minutes before removing to a wire rack to cool completely. Cookies can be stored in airtight containers in the freezer for up to 1 month.

Remove the ice cream from the freezer and assemble ice cream sandwiches. Freeze the sandwiches until firm, at least 1 hour, before serving.

Sandwiches can be individually wrapped in plastic and stored in the freezer for up to 1 week. Let sit at room temperature for 5 to 10 minutes before serving.