In a large bowl, cream together the softened unsalted butter and granulated sugar using an electric mixer on medium speed until light and fluffy. This should take about 2-3 minutes.

Beat in the large egg and vanilla extract until well combined. Scrape down the sides of the bowl as needed.

In a separate medium bowl, whisk together the all-purpose flour and salt.

Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix; stop when streaks of flour disappear.

Divide the dough in half. Shape each half into a disc, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes (or up to 2 days) to firm up. This makes the dough easier to handle.

Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper or silicone baking mats.

On a lightly floured surface, roll out one disc of dough to about 1/4-inch thickness. Use cookie cutters to create your desired shapes. Gather scraps, re-roll once, and cut.

Carefully place the cut-out cookies onto the prepared baking sheets, spacing them about 1 inch apart.

Bake for 8-10 minutes, or until the edges are lightly golden brown. The centers should still look slightly soft.

Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.