In a medium bowl, whisk the flour and salt. In a large bowl, using an electric mixer on medium-high, beat the butter and sugar until light and fluffy, about 2 minutes. Beat in the egg and vanilla. Add the flour mixture; beat on low speed until just blended. Shape into a disk, wrap in plastic and refrigerate until firm, about 1 1/2 hours.
Position the racks in the top and bottom thirds of the oven; preheat to 350 degrees . Roll 2 tsp. of dough into a 1-inch ball. Repeat with the remaining dough, dividing the balls between two parchment-lined baking sheets, spacing 1 inch apart. Using your thumb, make an indentation in the center of each dough ball.
Bake the cookies until firm but not browned, switching and rotating the sheets halfway through baking, 13 to 15 minutes. If the indentions puff up during baking, lightly press them down again with your thumb. Transfer the cookies to a rack and let cool completely. Store in an airtight container up to 1 week.