Beat the butter, sugar, vanilla, and salt:
In the bowl of a stand mixer with the paddle attachment or in a large mixing bowl with a hand mixer, beat together the softened butter, sugar, vanilla extract, and salt on medium-high speed until the mixture is pale and fluffy, about 2 minutes.
Add the flour:
Add the all-purpose flour to the butter mixture and mix on low speed until a stiff dough forms. The dough might seem dry at first, but keep beating until it comes together, stopping halfway through to scrape down the sides of the bowl with a rubber spatula.
Chill the dough:
For cut-out cookies: Place a large piece of plastic wrap on the counter. Scrape the dough onto the plastic wrap and shape it into a disk about 1-inch thick. Wrap the dough in the plastic and chill in the refrigerator until firm, at least 30 minutes.
For slice-and-bake cookies: Place a large piece of plastic wrap on the counter. Scrape the dough onto the plastic wrap and shape it into a log about 2 1/2 inches in diameter. Wrap the log in the plastic and chill in the refrigerator until firm, at least 1 hour.
Preheat the oven and prepare baking sheets:
Arrange racks in the lower and upper thirds of the oven. Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
Cut out the cookies:
For cut-out cookies: Dust your work surface generously with flour. If the dough is too stiff to roll, press a rolling pin firmly into the dough to make it more pliable. Roll the dough to 1/4 to 1/2 inch thick, rolling from the center outward. Cut out cookies using a 2 1/2-inch round cookie cutter. Space the cookies about 1 inch apart on the prepared baking sheets.
For slice-and-bake cookies: Use a sharp knife to slice the dough into cookies about 1/4 to 1/2 inch thick. Space the cookies about 1 inch apart on the prepared baking sheets.
Bake:
Bake the cookies until they start to turn lightly golden on the bottom and edges, about 20 minutes, rotating the pans halfway through. The middle of the cookies should look matte and feel dry to the touch. Depending on the thickness of the cookies, larger cookies may require an extra 5 minutes of baking time.
Cool and enjoy:
Allow the cookies to cool on the pans for 10 minutes before transferring them to a wire rack to cool completely. Enjoy your classic shortbread cookies with tea, coffee, or as a delightful treat on their own.