Preheat the oven to 325ºF (160ºC). Place the white sugar in a small container or bowl and have two baking sheets ready.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, brown sugar, vanilla, and salt on medium-high speed until well combined, about 4 to 5 minutes, scraping down the sides of the bowl halfway.
With the mixer running on low, add the flour gradually, a couple of tablespoons at a time. Wait until each addition of flour is mixed in before adding more. Stop the mixer and scrape down the paddle if the dough starts to clump around it. As more flour is added, the cookie dough may seem dry, however, it will eventually become smooth.
Once all of the flour is added, and the dough has pretty much cleared the sides of the bowl, stop the mixer. Dump the cookie dough out onto a work surface and knead it 15 to 20 times to ensure the flour is evenly distributed.
Divide the dough into approximately 24 portions (I like to use a scale to weigh out 25g portions). Use your hands to roll one portion into a smooth ball. Place the ball in the container of sugar. Swirl the ball gently inside the container to coat it in sugar. Repeat this process for all of the cookie dough balls, placing them on the prepared baking sheet(s) as you go, 12 balls per baking sheet.
Use a 2-¼ inch (5-½ cm) cookie stamp to press down and flatten each sugar-coated ball directly on the baking sheet. Press the stamp straight down, gently but firmly, and flatten the balls only to the edge of the stamp until the edges of the cookies just start to crack.
Once all of the cookies are stamped, place the two baking sheets in the preheated oven and bake the cookies for 22 to 24 minutes, rotating the pans halfway through baking. The cookies are done when the edges are lightly golden. Remove the sheets from the oven and place them on a wire rack to cool.