Cream together the butter and 1/2 cup confectioners' sugar until smooth in a bowl. Stir in the poppy seeds and vanilla extract. Stir in the flour until just combined. Roll the dough into 1-inch balls, and place 1 inch apart onto ungreased baking sheets.
Bake in the preheated oven until lightly golden, 8 to 10 minutes. Remove from baking sheet to cool slightly, then roll warm cookies in the remaining 1/2 cup confectioners' sugar until coated. Cool completely on wire racks, then roll in sugar once more.