In the bowl of a stand mixer fitted with the paddle attachment, beat butter, sugar, and salt at medium speed until smooth and creamy, 1 to 2 minutes, stopping to scrape sides of bowl. Beat in egg and vanilla until combined, stopping to scrape sides of bowl. With mixer on low speed, gradually add flour, beating just until combined.
Turn out dough onto a sheet of plastic wrap, and shape dough into a disk. Wrap in plastic wrap, and refrigerate until firm, at least 1 hour.
Preheat oven to 350°F (180°C). Line baking sheets with parchment paper.
On a lightly floured surface, roll dough to 3/16-inch thickness. Using a 2¼-inch flower-shaped cutter, cut dough. Using a 1-inch round cutter, cut out center from flower shapes. Place cookies 1 inch apart on prepared pans. Reroll scraps, and cut to use all dough.
Bake until edges are lightly browned, 18 to 22 minutes. Let cool on pans for 10 minutes. Remove from pans, and let cool completely on wire racks. Store in an airtight container for up to 1 week.