Using a stand mixer fitted with a paddle attachment or a hand mixer, cream the butter on medium speed until pale yellow, about 2 minutes. Scrape down the sides of the bowl and the paddle or beaters. Add the sugar, salt, and vanilla. Cream on medium speed until it is smooth about 1 minute. Stop the mixer and scrape down the sides of the bowl and the paddle.
Add the egg and beat on low speed for 15 seconds, or until fully incorporated. Do not overbeat. Scrape down the sides of the bowl and the paddle.
On low speed, add the flour. Beat until all of the flour is incorporated. Scrape down the sides of the bowl, then mix on low speed for 15 to 30 seconds, or until you have even-textured dough.
Remove small handfuls of dough from the mixer and plop them down the middle of a sheet of parchment paper, creating a log of 1 1/2 inches wide and 12 inches long. Fold the parchment over, creating a sausage. Chill for at least 1 hour. At this point, the dough will keep nicely, tightly wrapped, in the refrigerator for up to 1 week or in the freezer for up to 1 month. (Thaw frozen dough at room temperature for about 30 minutes, or until you can slice it.)
Preheat the oven to 350 degrees F. Adjust the rack to the lower third of the oven. Line two baking sheets with parchment paper.
When the dough has chilled, remove it from the parchment, and if using sprinkles, pour a 1/4 cup of sprinkles on to your work surface and roll the long in the sprinkles. Using a chef's knife slice 1/3-inch-thick rounds off the log. Place the cookies 1 inch apart on the prepared baking sheets. Alternate the rows in a checkerboard fashion.
Bake one sheet at a time 12 to 15 minutes, or until golden brown around the edges, turning the baking sheet from to back halfway through the baking. Remove from the oven and carefully slide the parchment onto a work surface. Wait at least 5 minutes before serving or 20 minutes before storing in an airtight container for up to 3 days at room temperature. The cookies can also be frozen for up to 1 month.
Notes
Adapted from from Sherry Yard, "Secrets of Baking"