To make the ice cream: Combine the milk, granulated sugar, and ginger in a medium saucepan and bring to a boil. Immediately reduce the heat and simmer until the sugar has fully dissolved, about 5 minutes. Remove from the heat and add the vanilla, heavy cream, and salt. Let cool, and then transfer the mixture to a container, cover, and refrigerate until thoroughly chilled, at least 4 hours and up to 24 hours.

Churn the mixture in an ice cream maker according to the manufacturer’s directions. Allow the ice cream to further harden, at least 2 hours, in the freezer before assembling the ice cream sandwiches.

To make the cookies: In a medium bowl, sift together the flour, baking soda, ginger, cinnamon, cardamom, and sea salt. Set aside.

In the bowl of a standing mixer fitted with the paddle attachment, combine the butter with the dark and granulated sugars on medium speed until light and fluffy. Turn the machine off, add the molasses, and then beat until fully incorporated. Add the egg and vanilla, scraping down the sides of the bowl as necessary, and continue to beat until blended. Reduce the mixer speed to low and add the flour mixture, one-third at a time, beating until blended. Transfer the dough to a piece of parchment paper, and roll it to form a 2-inch-wide cylinder. Refrigerate, covered with parchment, for at least 30 minutes and up to overnight.

Position a rack in the center of the oven, and preheat the oven to 350°F. Line two baking sheets with parchment paper.

Using a spice grinder, process the candied ginger until it is granulated (or finely mince it using your knife). Set aside.

Slice the dough into 1/4-inch-thick rounds, and place them on the prepared baking sheets. Sprinkle the cookies with the candied ginger. Bake, one sheet at a time, for 10 to 12 minutes. Remove the cookies from the oven, transfer them to a wire rack, and cool completely.

Sandwich 1/2 cup of the ice cream between 2 cookies. Repeat with the remaining ice cream and cookies.