Salted Brown Butter Cookies

Melt butter in a 1- to 2-quart heavy saucepan over medium-low heat, then cook over low heat, swirling pan occasionally, until butter has a nutty aroma and is a rich brown color, and bottom of pan is speckled with browned bits of the milk solids, 8 to 10 minutes. Transfer to a bowl, including browned milk solids on bottom, and chill until almost firm, about 30 minutes. Whisk together flour and cornstarch in a bowl. Beat together butter and sugar in a large bowl with an electric mixer at medium-high speed until pale and fluffy, then beat in egg, salt, and vanilla. Add flour mixture and mix on low speed until just combined well. Form dough into 2 balls and flatten each into a 6-inch disk. Chill disks, wrapped in plastic wrap, until firmer, but not hard, 15 to 30 minutes. Heat oven to 350 °F with rack in middle position. Line a large baking sheet with parchment paper. Roll out 1 disk of dough (keep remaining dough chilled) between 2 large sheets of wax paper 1/8 inch thick (this is when the rolling pin rings are really helpful). Freeze or chill dough on a baking sheet or tray until firm, 5 to 15 minutes, then cut out as many cookies as possible from dough. Transfer cookies to lined baking sheet, arranging cookies about 1 inch apart. If the cookies become too warm to transfer, freeze or chill until firm (see Cooks' notes). Bake cookies, 1 sheet at a time, until edges are golden, 8 to 10 minutes, then transfer to racks to cool completely. Meanwhile, gather scraps of dough and repeat cookie process above. Parchment can be reused, but make sure baking sheet cools between batches. Make more cookies with remaining dough.