Preheat the oven to 300°F. Line a large baking sheet with parchment paper.
Beat the butter and sugar with an electric mixer until light and creamy.
In a small bowl or custard cup, combine the vanilla, baker's ammonia, and salt. Stir to dissolve the ammonia. Scrape vanilla mixture into butter mixture and beat until smooth.
Mix in the flour and stir until the dough holds together.
Using a small cookie scoop (or ½ ounce weight) of dough, shape into balls. Roll the balls in coarse sugar. Press down flat using the bottom of a glass or a spatula with holes in it.
For more decorative cookies with circular ridges, use the presser from a food processor to press the cookies (King Arthur's idea!)
Bake the cookies at 300 degrees until they're a very light golden brown around the edges, about 25 minutes.