Use stand mixer (e.g. Kitchenaid) to mix sugar, salt, and vanilla; then butter. Once that's thoroughly mixed, incorporate the flour. (Set aside butter wrappers to grease pans, if desired.)
Take the ball of dough and flatten it into a thick disk. Wrap it in plastic, or use a large plastic zip bag. Or just cover the dough in the mixer's bowl with plastic wrap. Chill it for 30 minutes.
When you take the dough out, heat the oven to 325° F. Line pans with baking parchment. Or use the butter wrappers to grease the pans.
Use small purple disher to scoop balls of dough, about the size of a walnut. Roll them in sugar. Place them 2" apart.
Pick a glass with an interesting indentation pattern on the bottom. (I use my crystal creamer or sugar bowl.) Use it to flatten the cookies to a uniform thickness, dipping the glass in the sugar before every other cookie.
Bake at 325° F for 14-16 minutes or just until the edges start to brown. (Use the shorter time for dark pans.)
Take out of the oven. Let them cool on the pan for two minutes, then transfer to wire rack.
[Sep. 2023. I got 44 cookies. 16 minutes is probably right. (I tried 20, 18, and 16 minutes.) I combined two similar recipes. One of them called for 3 cups of flour, just a pinch of salt, a 325°F oven, and 30-35 minutes baking time. The other wanted 3½ cups, ¼ tsp salt, and 350°F, which are what I used; and 20-25 minutes. Interestingly enough, both recipes wanted you to preheat the oven before you put the dough in the fridge for half an hour...]
[Oct. 2023. I got 45 cookies. My darker pans bake faster. I've changed the recipe to 3 cups/325°F, and bumped up the vanilla up 50%. The first batch with 3½ cups of flour, I scooped them; and on this batch, I used the spoon-and-level method for my 3 cups. It doesn't seem to make much difference.]
[Oct. 2023, later. I got 48 cookies. My shiny new pans need 20 minutes.]
[Dec. 2023. Spread out in dark 12x17 pan for 2"x2" bar cookies. Bake 20 minutes.]
[Feb. 2024. Spread out in dark 9x13 pan. Bake 26 minutes. 48 pieces, ¾" thick. Later: I think that was 26 minutes at 350°. 26 minutes at 325° gives a light buttery melt-in-your-mouth result.]