Line a quarter sheet pan or 9-by-13-inch baking pan with plastic wrap, leaving an overhang. Spread the ice cream in the pan in an even layer with an offset spatula. Freeze until very firm, about 4 hours.
Sift the flour, baking powder, cinnamon, ginger, allspice and cloves into a large bowl. Beat the butter and brown sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 4 minutes. Add the eggs one at a time, beating after each addition until smooth. Reduce the mixer speed to low and beat in the flour mixture until just combined. (If the dough seems dry, add 1 to 2 tablespoons water.) Transfer the dough to a sheet of plastic wrap, shape into a disk and wrap tightly. Refrigerate until chilled, about 20 minutes.
Line 3 baking sheets with parchment paper. Roll out the dough on a lightly floured surface until about 1/8 inch thick. Cut the dough into 2-by-4-inch rectangles (a pizza cutter is great for this); you’ll get about 24. Arrange on the prepared pans, about 1 1/2 inches apart, and prick each cookie with a fork a few times down the center. Freeze until firm, 15 to 30 minutes.
Preheat the oven to 325 degrees F. Bake the cookies, one baking sheet at a time, until golden on the bottom but still soft in the center, 15 to 20 minutes. Transfer to a rack and let cool 5 minutes on the pan, then transfer the cookies to the rack to cool completely.
Lift the ice cream out of the pan using the plastic wrap and invert onto a cutting board; discard the plastic. Working quickly, cut the ice cream into twelve 2-by-4-inch rectangles. Sandwich the ice cream between the cookies, place on a baking sheet and return to the freezer.
Put the white chocolate in a microwave-safe bowl and microwave in 30-second intervals, stirring, until smooth. Stir in the vegetable oil. Dip each sandwich partway in the white chocolate, then sprinkle with nonpareils. Return to the baking sheet and freeze until ready to serve.