Whisk the vanilla and egg in a small bowl and set aside.
In a large bowl with a hand-held mixer, cream the butter until fluffy. Scrape down the sides of the bowl with a rubber spatula, add the coconut sugar and continue beating until light, 2 to 3 minutes. Scrape the sides of the bowl again if needed, then add the egg mixture, beating for about 1 minute. Sift the flour with the salt in a small bowl. Reduce the mixer speed to low, then add the flour mixture, mixing just until blended.
Turn the dough out of the bowl. Divide the dough in half and place each half between 2 pieces of lightly floured parchment or waxed paper. Use a rolling pin to flatten the dough into a disk about 1/4 inch thick. Slide the dough/parchment sheet onto a flat baking sheet or on the back of a rimmed baking sheet. Refrigerate until firm, about 2 hours and up to overnight.
Position oven racks in the top and bottom thirds of the oven and preheat to 325 degrees F. Line 2 baking sheets with parchment (or use nonstick baking sheets).
Place 1 disk of the cookie dough on a clean work surface and peel off the top sheet of paper. Cut the cookies with a 1 1/2-inch round or decorative-shaped cookie cutter directly on the parchment. Use a spatula to transfer the cookies to the prepared baking sheets, spacing them about 1 inch apart. Repeat with the other sheet of dough. Any excess dough can be re-rolled, refrigerated and cut.
Bake until the edges of the cookies are light golden, about 15 minutes. Transfer the cookies to a rack to cool.