How to Make It
Prepare an ice bath in a large bowl; set aside. Melt butter in a medium-size deep saucepan over medium, swirling often, until butter bubbles and develops a heavy foam, foam subsides, and butter turns a deep golden color, about 14 minutes. Remove from heat; pour brown butter into a heatproof bowl, scraping milk solids from bottom of saucepan into bowl. Place bowl in prepared ice bath; let cool, stirring occasionally, 10 minutes. Remove bowl from ice bath; let stand at room temperature 2 hours. Cover and chill overnight or up to 3 days.
Remove brown butter from fridge; let stand at room temperature 1 to 2 hours to soften. Transfer butter to bowl of a heavy-duty electric stand mixer fitted with paddle attachment. Add 1 cup granulated sugar and powdered sugar. Beat on medium-low speed until smooth and creamy but not yet light and fluffy, about 2 minutes. (Do not overbeat.) Add egg yolks, 1 at a time, beating after each addition, stopping to scrape down sides of bowl as needed. Add vanilla, and beat until just combined, about 5 seconds. Reduce speed to low; gradually add flour and kosher salt, beating until just combined, about 1 minute.
Form dough into a ball; transfer to a large sheet of parchment paper. Cover with a second sheet of parchment paper. Using a rolling pin, gently roll dough between parchment to 1/3-inch thickness. Transfer dough sandwiched between parchment paper onto a baking sheet. Chill until firm, 1 hour or overnight.
Remove top sheet of parchment paper from chilled dough. Cut dough into 24 (3-inch) circles. (Reroll dough scraps as needed; rechill dough if it becomes too soft to handle.) Transfer dough rounds to a baking sheet lined with parchment paper; freeze until solid, about 1 hour.
Preheat oven to 350°F. Remove 12 dough rounds from freezer. Place remaining 1/2 cup granulated sugar in a small bowl. Brush edges of the 12 dough rounds with half of the egg white; roll edges of dough rounds in granulated sugar to coat. Place dough rounds 3 inches apart on 2 baking sheets lined with parchment paper. Sprinkle tops evenly with half of the sel gris. Bake in preheated oven until centers of cookies are just set and edges are lightly browned, about 14 minutes. Remove from oven. Cool cookies on baking sheets 2 minutes; transfer to a wire rack to cool completely, about 30 minutes. Repeat procedure with remaining 12 dough rounds, remaining half of egg white, remaining granulated sugar (discard leftover granulated sugar once all 24 cookies have been rolled), and remaining half of sel gris.