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Cream the butter and sugar. Cut 2 sticks cold unsalted butter into 1/2-inch cubes. Place the butter and 1/2 cup granulated sugar in a stand mixer fitted with the paddle attachment. (Alternatively, use a large bowl and electric hand mixer.) Beat on low speed to break up the butter, then increase the speed to medium and beat until fluffy and lightened in color, about 3 minutes.
Add the egg and vanilla. Stop the mixer and scrape down the sides of the bowl with a rubber spatula. Add 1 large egg and 1 teaspoon vanilla extract, and beat on medium speed until combined.
Beat in the flour and salt. With the mixer on low speed, slowly add 2 cups all-purpose flour and 1/4 teaspoon kosher salt, and beat until just incorporated. Use a rubber spatula to scrape down the sides of the bowl and make sure the dough is evenly mixed.
Roll and chill the dough. Remove the dough from the bowl and shape into a smooth rectangle. Lightly dust the rectangle with flour. Place between 2 sheets of parchment paper, then roll into a rough 9x12-inch rectangle about 1/4-inch thick. Place the dough, still in the parchment, on a baking sheet. Freeze until firm, 20 to 30 minutes, or refrigerate 1 to 2 hours. Meanwhile, heat the oven and prepare the egg wash.
Make the egg wash and heat the oven to 350ºF. Whisk the remaining 1 large egg with 1 teaspoon water together in a small bowl. Arrange 2 racks to divide the oven into thirds and heat the oven to 350°F.
Cut the cookies. Remove the top sheet of parchment from the dough. Using a pizza cutter or knife, cut the dough into 36 (1.5x2-inch) rectangles. Arrange the cookies 1-inch apart on 2 ungreased, unlined aluminum baking sheets (not nonstick), about 18 cookies per sheet.
Finish the cookies with egg wash and sugar. Brush the top of the cookies with the egg wash and sprinkle with 1/4 cup sanding or coarse sugar.
Bake the cookies 10 to 14 minutes. Bake the cookies until firm and lightly golden-brown, rotating the baking sheets between racks and from front to back halfway through, 10 to 14 minutes total.
Cool on the baking sheets for 5 minutes. Let the cookies cool for 5 minutes on the baking sheets. Transfer to a wire rack to cool completely.