Preheat oven to 350°F. Line two baking sheets with parchment.
Whisk flour, baking powder, baking soda, and salt in a bowl.
Cream butter and granulated sugar until pale and fluffy, about 2 minutes.
Beat in beaten egg, lemon zest, lemon juice, and vanilla until smooth.
Mix in dry ingredients on low just until a soft dough forms. Do not overmix.
Scoop tablespoon-size mounds onto sheets, spacing about 2 inches apart.
Bake 10–12 minutes, until edges are set and tops look matte. Cool 5 minutes, then transfer to a rack.
Whisk confectioners' sugar, lemon juice, milk, and zest until silky. Adjust with a splash of milk if thick.
Glaze cooled cookies by dipping tops or drizzling. Let set until the glaze is firm.