Place 2 sticks unsalted butter in the bowl of a stand mixer (or large bowl if using an electric hand mixer). Let sit at room temperature until softened. Meanwhile, line an 8- or 9-inch square metal baking pan with a sheet of parchment paper that extends over two opposite sides.
Beat the butter with the paddle attachment on high speed until fluffy, 3 to 5 minutes. With the mixer running on low speed, gradually add ¾ cup powdered sugar and continue to beat until pale and fluffy. Scrape down the sides of the bowl with a flexible spatula.
Sift 2 cups all-purpose flour into the bowl, then add 1 teaspoon vanilla extract and ¼ teaspoon coarse salt. Mix on low speed until just combined. Transfer to the pan and pat to an even thickness no more than ½-inch thick. Refrigerate for at least 30 minutes. Meanwhile, arrange a rack in the middle of the oven and heat the oven to 300°F.
Cut the dough into 16 pieces in the pan. Bake until light golden-brown and set in the middle, 45 to 60 minutes. Let cool completely. Grasping the excess parchment paper, lift the slab out of the pan. Re-slice, if needed, before serving.
