In the bowl of a standing electric mixer fitted with a paddle attachment, cream butter and 1/3 cup of the sugar until very light and fluffy, about 4 minutes.
Turn off the mixer and add the vanilla extract, then beat to combine.
Turn off the mixer, add the flour and salt, and mix on low speed until the flour is just incorporated. Turn off the mixer.
Using a rubber spatula, transfer the dough to a piece of plastic wrap and wrap completely. Refrigerate for 1 hour, or until the dough is cool to the touch and mostly firm.
Position one rack in the center and another in the lower third of the oven and preheat the oven to 375ºF.
Remove the dough from the refrigerator and, using your hands, make 18 walnut-sized balls of dough, about 2 tablespoons each. Divide these evenly between the baking sheets, 3 inches apart, and press down on each cookie to form a slightly flattened disk.
Sprinkle 1/4 teaspoon of the sugar evenly over the top of each cookie.
Using oven mitts or pot holders, transfer the baking sheets to the oven and bake until cookies are golden brown around the edges, about 16 to 18 minutes, exchanging positions of baking sheets after 8 minutes.
Using oven mitts or pot holders, remove the baking sheets from the oven and set on wire racks to cool completely before removing the cookies from the baking sheets with a metal spatula. Be careful -- these cookies are fragile while they're warm.
Cookies will keep for several days if kept in an airtight container.
YIELD: Eighteen 4-inch cookies