Preheat oven to 350 degrees. Place rack in middle of oven. Line two large baking sheets with parchment paper.
In the bowl of a stand mixer fitted with paddle attachment, cream butter and confectioners' sugar together until light and fluffy, about 2 minutes. Add lemon zest and vanilla and beat until combined.
Add the flour, poppyseeds and salt and beat until just combined. Do not overmix. Form the dough into a disk and wrap in plastic wrap. Place in refrigerator for 15 minutes.
Remove from fridge and place on lightly floured surface. Roll dough out to about ¼-inch thickness. Cut with 2-inch round cookie cutter. Transfer to prepared sheets placing 1 inch apart. Place sheets in refrigerator for 15 minutes.
Remove from refrigerator and bake, one sheet at a time, for 12-14 minutes or until lightly browned around the edges. Let cool on sheet for 5 minutes, then remove to wire rack over a piece of parchment to cool completely. Repeat with remaining dough until you've used it all.
