In a stand mixer fitted with the whisk attachment, beat together the butter and sugar on high speed until creamy, about two minutes. Add three of the eggs, one at a time, beating on high speed after each addition until combined. Add the vanilla and beat until combined, about two minutes longer.

Switch to the paddle attachment and add salt, and then add the flour, half a cup at a time, beating on medium speed until the dough is soft and holds together without being sticky. Add the flour gradually to achieve a smooth dough (you may not need all of it). Stir in the sprinkles.

Divide the dough into two equal balls and wrap separately in plastic wrap. Place in the fridge for at least one hour or up to three days.

Preheat the oven to 350 degrees Fahrenheit (180 degrees Celsius).

Place a piece of Gefen Parchment Paper that is the same size as a baking sheet on your counter and place a ball of the dough on the parchment. Place a second piece of parchment over the dough. Roll out the dough a quarter inch (six millimeters) thick.

Peel off the top parchment and use one or more cookie cutters to cut out shapes, spacing them one inch (two and a half centimeters) apart. Alternatively, use an overturned glass to cut out rounds. Keep the cookies on the parchment and pull away the excess dough.

Shape the excess dough into a ball, place between parchment sheets, roll out a quarter inch (six millimeters) thick, and cut out more cookies.

Transfer the parchment paper with the cookies to a baking sheet. In a small bowl, beat the remaining egg, then brush the cookies with it.

Bake the cookies until golden, 10–15 minutes. Let cool on the baking sheet for 15 minutes, then transfer to a wire rack to cool completely. While the first pan of cookies is baking, repeat with the second ball of dough to make more cookies. Bake a third batch of cookies from any excess dough scraps.