Danish Salted-Butter Cookies
['1 cup (2 sticks) chilled salted butter, cut into pieces', '3/4 cup granulated sugar', '1 vanilla bean, split lengthwise', '2 large eggs', '2 cups all-purpose flour, plus more', '1 cup white sanding sugar', 'A 3"-diameter and a 1 1/2"-diameter fluted cookie cutter']
Combine butter and granulated sugar in a large bowl; scrape in vanilla seeds and discard pod. Using an electric mixer on high speed, beat until very light and fluffy, about 5 minutes. Beat in 1 egg. Reduce speed to low; gradually mix in 2 cups flour. Form dough into a 3/4"-thick disk. Wrap in plastic and chill until firm, about 2 hours.
Place racks in upper and lower thirds of oven and preheat to 350°F. Let disk of dough sit at room temperature until slightly softened, about 5 minutes. Roll out on a lightly floured sheet of parchment paper, shifting often on sheet and dusting with more flour as needed to keep from sticking, until about 3/8" thick.
Punch out rounds with large cutter; punch out centers with smaller cutter. Transfer to parchment-lined baking sheets, spacing 1 1/2" apart. Pat scraps into a 3/4"-thick disk. Chill 15 minutes.
Beat remaining egg in a small bowl with 1 tsp. water; brush cookies with egg wash; sprinkle with sanding sugar. Freeze 10 minutes. Bake, rotating baking sheets halfway through, until edges are golden brown, 10–15 minutes. Transfer to wire racks; let cool. Repeat with scraps and remaining egg wash and sanding sugar.
Dough can be made 1 month ahead; freeze. Cookies can be baked 2 days ahead; store airtight at room temperature.
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