Position oven racks in upper and lower thirds of oven; preheat to 375°F. Line 2 large rimmed baking sheets with parchment paper.
Whisk 1 cup all-purpose flour, 1 cup whole-wheat flour, ½ cup granulated sugar, 1 tablespoon baking powder and ½ teaspoon salt together in a large bowl until combined.
Add ¼ cup butter and beat with an electric mixer on low speed until the butter is completely blended into the flour mixture, about 1 minute.
Add ½ cup milk, 1 egg, 2 tablespoons lemon zest and 1 teaspoon vanilla; continue beating on low speed until blended, about 1 minute, scraping down sides of bowl as needed.
Chill the dough, uncovered, for 30 minutes.
Using lightly floured hands, roll the dough into 24 (1-inch) balls and place 1 inch apart on the prepared baking sheets.
Bake on the upper and lower oven racks until the edges are light golden and the tops are dry to the touch, 10 to 12 minutes.
Cool on the pans for 5 minutes before transferring to wire racks to cool completely, about 30 minutes.
Whisk 2 cups confectioners' sugar and ¼ cup lemon juice together in a small bowl until smooth.
Place the cooled cookies, still on the wire racks, over the parchment paper-lined baking sheets.
Drizzle the glaze (about 2 teaspoons per cookie) over the tops of the cookies.
