In the bowl of an electric mixer fitted with the paddle attachment, combine butter, salt and vanilla; beat on high speed until fluffy, 2 to 3 minutes. Separate the egg yolks from the whites, and place the yolks in a medium sized bowl, reserving the whites for later.
Whisk the egg yolks by hand with the sugar until fluffy, about 1 minute. Add the egg mixture to the butter and beat on low to combine. Sift the flour and baking powder together, then add to the butter mixture, and mix on low speed until flour is incorporated. Do not overmix.
Roll dough into three 1 1/2-inch (about 4 cm) diameter logs. Wrap in plastic wrap, and refrigerate until firm, at least 1 hour.
Heat oven to 350 degrees F (175C) with a rack in the middle of the oven. Line a baking sheet with parchment paper or a Silpat. Slice the dough into 1/4-inch (0.6 cm) rounds. Place cookies on baking sheet about 1 inch (2.5 cm) apart. Brush the tops of the cookies with the egg white, and bake for 8-10 minutes. Continue with the rest of the dough. Allow baked cookies to cool on a wire rack, and enjoy.
Dough can be chilled for up to 3 days in a sealed container or frozen for 1 month. Once baked, the cookies will keep for four days in an airtight container.