Heat oven to 350° F and line two baking sheets with parchment paper. In a medium bowl, whisk together the flour, baking powder, and salt. In a separate, large bowl, beat the butter and sugar on medium-high speed until fluffy, 2 to 3 minutes. Add the eggs one at a time, followed by 2 teaspoons vanilla extract and the seeds from ½ vanilla bean. Reduce the speed to low and gradually add the flour mixture until just combined.
Roll the cookie dough into tablespoon-sized balls and place them on the prepared baking sheets with 2 inches of space between them. Bake until pale golden at the edges, about 12 minutes. Cool the cookies on the baking sheets for 2 minutes, then transfer them to a wire rack to cool completely.
Make the glaze: In a small bowl, whisk together the confectioners' sugar, milk, ½ teaspoon vanilla extract, and the seeds from 1 vanilla bean. The consistency should be thick but pourable (add a splash more milk, if necessary). Dip the top of each cookie into the glaze and let set, rounded side up, on a rack set over a piece of parchment paper.
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