In the bowl of an electric mixer fitted with a paddle attachment, beat the butter, sugar, and lemon zest on high until light and creamy, 2-3 minutes. Mix in the vanilla.
Add the flour, salt and poppy seeds and mix until it comes together into a stiff dough.
Shape the dough into a log 2-3 inches in diameter and cover with plastic wrap. Chill in the fridge for at least 1 hour and up to 5 days.
When ready to bake, preheat oven to 350°F and line sheet pans with parchment paper.
Slice cookie dough into 1/2-inch thick slices and place on sheet pans a few inches apart.
Bake for 10-12 minutes, just until the bottoms start browning. Let cool on the sheet pan a few minutes, then transfer to a wire rack to cool completely.
To make the glaze, whisk together the powdered sugar and lemon juice. Drizzle over the cooled cookie and let glaze set up for 15 minutes before serving.