For the ice cream: In a large saucepan, bring the cream, milk, half of the granulated sugar, the fresh ginger and candied ginger to a boil. Remove from the heat and steep for 10 to 15 minutes. Bring the mixture back to a boil.
Meanwhile, whisk the yolks and remaining sugar until light and pale. Gradually add the hot dairy mixture to the yolks, whisking constantly. Continue tempering until the dairy mixture (including the solids) is fully incorporated.
Dump the mixture back into the saucepan and cook, stirring constantly over medium heat, until thick and the mixture can coat the back of a spoon, 2 to 3 minutes.
Strain the mixture through a fine mesh sieve into a clean bowl set over an ice bath and discard the solids. Stir frequently until mixture cools completely, about 1 hour.
Cover and chill the ice cream base in the refrigerator for at least 8 hours.
Process the chilled ice cream base in an ice cream maker according to the manufacturer's instructions. Freeze the ice cream in the freezer until firm, at least 2 hours or up to overnight.
For the cookies: Preheat the oven to 350 degrees F. Line a small baking sheet with plastic wrap. Line two baking sheets with parchment paper.
In a stand mixer fitted with the paddle attachment, mix the butter at medium speed until light and smooth, about 3 minutes. Add the granulated sugar and continue to mix until light and fluffy, about 5 minutes. Add the eggs and salt, and mix until incorporated.
In a measuring cup, mix together the molasses and vinegar, then add to the mixer and mix at medium speed until incorporated. Add the candied ginger and fresh ginger and mix until fully incorporated, about 2 minutes.
Whisk together the flour, baking soda, ground ginger, cardamom, cinnamon, cloves and allspice. With the mixer running on the lowest speed, add the dry ingredients, and mix until combined.
Place the dough on the plastic-lined baking sheet, and wrap tightly with plastic wrap. Chill the dough until firm, about 30 minutes.
Scoop the cookie dough with a 2-tablespoon scoop. Roll half of the dough balls in granulated sugar and roll the other half in turbinado sugar. Place the dough balls on the parchment-lined baking sheet (about 48 cookies in total) and bake until the cookies are just set, 8 to 12 minutes. The cookies will crackle on top.
Pull the cookies from the oven and firmly tap the baking sheet on a solid surface to deflate the cookies. Cool completely on the sheet tray.
Scoop some ice cream and sandwich between a granulated-sugar-coated cookie and a turbinado-sugar-coated cookie. Repeat for the remaining cookies and ice cream. Serve immediately or store the ice cream sandwiches in the freezer until ready to eat.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.