In a large mixing bowl, cream the softened butter and sugar together using an electric mixer on medium speed until light and fluffy, about 3-4 minutes. This step is crucial for achieving that tender, crumbly texture.
Crack in the egg and pour in the vanilla extract. Beat the mixture until everything is well combined, scraping down the sides of the bowl as needed.
In a separate bowl, whisk together the flour and salt. Slowly add this dry mixture into the butter mixture, mixing on low speed until a smooth dough forms. Avoid overmixing, as this can lead to tougher cookies.
Shape the dough into a disc and wrap it in plastic wrap. Refrigerate for at least 30 minutes. This step is important because chilled dough is easier to handle and results in less spreading during baking.
Preheat your oven to 350°F (175°C) and line your baking sheets with parchment paper or a silicone baking mat.
Once the dough has chilled, roll it out on a lightly floured surface to about ¼-inch thickness. Use a cookie cutter to cut out your desired shapes, or simply roll the dough into small balls and flatten them gently with a fork for a classic look.
Place the cookies about 1 inch apart on the prepared baking sheets. Bake for 12-15 minutes, or until the edges are lightly golden. Keep an eye on them to prevent overbaking, as butter cookies can quickly turn from golden to burnt.
Remove the cookies from the oven and let them cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely. This helps them set without breaking apart.