With a hand mixer on low speed, blend butter until creamy. Add egg and continue mixing until well blended. Slowly add the dry gingerbread cookie mix and continue blending on low until a dough starts to form. Add cold water, one tablespoon at a time until the dough becomes moist and begins to pull away from the bowl. Divide the dough in half and roll each portion out onto a piece of parchment paper until the dough is 1/4" thick. Stack the dough and parchment paper onto a sheet pan and freeze for 15 minutes or until firm.
Preheat the oven to 350 degrees and spray the popover pan with non-stick cooking spray. Remove one of the dough sheets from the freezer and use a small circle cookie cutter to cut dough for the bottom of the cookie shot glasses. If you do not have a small circle cookie cutter, you can use the rim of a small drinking class to cut the dough. Remove the second sheet of dough from the freezer. To create the sides of the cookie shot, use a knife to cut strips of dough that are slightly taller than the height of the popover pan cups.
Place a cut-out circle of dough into the bottom of each popover cup, gently pressing it down and slightly up the sides. Next, press the dough strips into the cups along the sides, slightly overlapping the bottom dough to create a seal. Continue adding strips and gently pressing them around the sides of each popover cup.
Spray six shot glasses with non-stick spray and place one into each well of dough. This will help keep the dough from rising and will maintain the shape of each cookie shot. If you do not have shot glasses, use wine corks covered with aluminum foil and spray with non-stick spray.
Bake for 17-20 minutes, until the top edges begin to brown and the bottom of the cookie shot is baked and set. Remove from oven and place the popover pan on a wire rack to cool. Once cooled, use a butter knife to gently remove the cookie shots from the pan. Cool completely.
Place the white chocolate in a microwave safe bowl and heat in 15 second increments, stirring after each time, until completely melted and smooth. Working with one cookie shot at a time, pour a tablespoon of melted chocolate into the bottom and immediately swirl the cookie shot to evenly distribute the chocolate. Repeat with each cookie shot, then place them on a baking sheet and freeze for 5 minutes to set the chocolate. Fill each cookie shot up with egg nog and serve immediately.