Cream together the butter and sugar in a medium bowl until light and fluffy, about 3 minutes.
Mix in the vanilla extract and egg until well incorporated.
Gradually add the flour, combining until just mixed. Do not over-mix.
Stir in the poppy seeds and lemon zest until evenly distributed throughout the dough.
Divide the dough into two portions and shape into logs, about 6 inches long and 1.5 inches in diameter. Wrap in plastic and refrigerate for at least 1 hour.
Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper.
Slice the dough logs into 1/4-inch rounds and place on prepared baking sheets. Bake for 10-12 minutes or until edges are golden.
Notes
To make gluten-free, replace all-purpose flour with a gluten-free flour blend. Ensure all other ingredients are certified gluten-free. Store cookies in an airtight container for up to a week.