Pour pink champagne into a medium saucepan over high heat. Boil champagne until reduced to 3/4 cup, about 15 to 20 minutes. Transfer to a measuring cup; chill until cold.

In a large bowl, whisk together flour and baking powder.

In a separate large bowl or bowl of a stand mixer, cream 1/2 cup sugar and shortening until smooth. Add 1/2 cup of the cold champagne reduction; stir to combine (shortening will be lumpy). Add flour mixture and stir until a very soft dough forms.

Heat oven to 375 degrees F. Line two baking sheets with parchment paper. Scoop dough into 1-inch balls, rolling each ball of dough in remaining sugar before transferring to prepared baking sheet.

Lightly press down dough with fingers or the back of a measuring cup dipped in sugar. Bake 12 to 15 minutes until edges are just set. Cool 2 minutes on baking sheets, then transfer to a cooling rack to cool completely.

Meanwhile, make the icing: Whisk together powdered sugar with remaining 1/4 cup cold champagne reduction and pinch of salt. Dip cooled cookies in icing, letting excess drip off. Let icing set for a couple of minutes, then sprinkle with pearl sugar or sprinkles. Let icing set completely before serving.