Instructions

1. Make the ice cream: In a small bowl, whisk together the sugar, dextrose, malted milk powder, and guar gum (if using). To a medium pot, add the milk and heavy cream, then whisk in the sugar mixture. Bring to a rolling boil over medium-high heat, then turn down the heat and simmer, stirring occasionally, until the sugar has dissolved and the mixture has thickened, about 5 minutes. Remove from the heat, then pour into a large heatproof bowl, cover, and refrigerate or freeze until cool to the touch, about 2 hours in the fridge, or 20 minutes in the freezer.
2. Uncover the ice cream base, sift in the matcha powder, and use an immersion blender to incorporate. Alternatively, add the matcha to a small bowl, whisk in ½ cup of the ice cream base to make a smooth paste, then whisk the matcha paste back into the bowl of ice cream base. (If any clumps remain, strain the liquid through a fine mesh sieve.) Cover and return the liquid to the fridge to rest for 8–24 hours.
3. Meanwhile, make the cookies: In a large bowl, whisk together the butter, brown sugar, egg, vanilla, ginger, salt, cardamom, and cinnamon until smooth. In a medium bowl, whisk together the flour and baking soda, then using a silicone spatula, fold the flour mixture into the butter-and-egg mixture. Refrigerate until the dough is cool but still soft and pliable, at least 20 minutes or up to 12 hours. (If refrigerating for more than an hour, cover the bowl.)
4. Position racks in the top and bottom thirds of the oven and preheat to 350°F. Scoop the dough into 28 two-tablespoon portions. Stagger the portions on two parchment-lined baking sheets, leaving at least 2 inches between them. (If the cookies do not all fit on two sheets, refrigerate the remaining dough and bake in batches.)
5. Bake, rotating and swapping the top and bottom pans halfway through cooking, until light golden brown but slightly underbaked in the center, 7–8 minutes. Set aside to cool completely on the baking sheets; the cookies will flatten and continue cooking as they cool. (If making ahead of time, transfer the cooled cookies to an airtight container and store at room temperature for up to 3 days.)
6. Transfer the matcha liquid to an ice cream machine and churn according to the manufacturer’s instructions until thick, frozen, and no longer wet in appearance, about 25 minutes. Add the salt and continue churning to incorporate, about 3 minutes.
7. Assemble the ice cream sandwiches: Line a baking sheet or large platter with parchment paper and place 14 of the cookies on it, bottoms up. Top each with approximately ¼ cup of ice cream, then top with the remaining cookies, pressing down slightly. (If you like, smooth the sides with the edge of an offset spatula.) Freeze until firm, at least 1 hour. Serve straight from the freezer. (Wrapped tightly in plastic, the sandwiches keep in the freezer for up to 2 weeks.)