Rub lemon zest into the 3/4 cup sugar with clean fingertips until fragrant. Beat with the butter until creamy, then beat in egg, lemon juice and vanilla.
Whisk flour, baking powder and salt and mix into the butter just until combined. Cover and refrigerate for 1 hour, additional to active prep time.
Heat the oven to 350°F and line two baking sheets. Divide the chilled dough into 24 balls; roll each in coating granulated sugar, then powdered sugar.
Bake one sheet at a time for 11 to 12 minutes, until cracked on top and set around the rim without dark browning.
Leave on the sheets for 5 minutes, then cool on racks before stacking.