In the bowl of stand mixer fitted with the paddle attachment, add the room temperature butter, sugar and lemon zest. Cream the mixture together on medium speed until fluffy and light in color, about 4 minutes. Scrape down the sides of the bowl as needed throughout. Beat in the egg until well incorporated, about 1 minute.
With the machine set to low speed, add the flour mixture, 1/2 cup at a time and mix until just combined.
Using a 2 tbsp cookie scoop, portion the dough onto the parchment lined sheet trays. Roll each ball with the palm of your hand, then space each about 2 inches apart on the sheet tray, gently press to flatten each ball to about 1/2 inch thick.
Bake on the middle rack of the oven for 15-20 minutes, until cookies are set and light golden around the edges. Cool cookies on the baking sheet for 5 minutes then transfer to a wire rack to cool completely.
Make the glaze, in a medium bowl, stir together the powdered sugar, lemon juice, and zest. Drizzle over the cookies and sprinkle with poppy seeds.