Make cookies: Using a stand mixer with paddle attachment, beat butter, sugar, and salt on medium speed until smooth. Beat in egg yolks and vanilla until incorporated. Stir or beat in 21/2 cups flour until blended. Flatten dough into a 1-in.-thick disk.
Preheat oven to 300° and line 2 rimmed baking sheets with parchment paper. On a lightly floured work surface, using a lightly floured rolling pin, roll dough to about 1/8 in. thick. Cut with cookie cutters and transfer to baking sheets, spacing about 1 in. apart (cookies do not spread). Bake, rotating pans halfway through, until golden, 15 to 20 minutes. Transfer cookies to wire racks to cool.
Reroll and cut dough scraps, reflouring work surface and rolling pin as needed; bake.
Make icing: In a clean mixer bowl using whisk attachment, combine powdered sugar and egg whites on low speed until moistened. Then beat on high until stiff, straight, glossy peaks form when whisk is lifted, about 15 minutes.
Spoon icing into a piping bag fitted with a plain 1/8-in.-wide tip. Pour a thick layer of sparkling sugar into a small dish. Pipe designs onto 1 cookie at a time and gently dip the still-wet piping into sugar (or carefully sprinkle sugar onto piping); tap off excess. Let dry on a wire rack until icing is firm, at least 1 hour.
MAKE AHEAD Dough, up to 2 days, chilled airtight (let stand at room temperature until soft enough to roll); or 1 month, frozen. Cookies, up to 2 days, stored airtight with sheets of waxed paper between layers; or 1 month, frozen.