Preheat the oven to 350°F. Line two baking sheets with parchment paper and set aside.

In a large bowl, add the granulated sugar and lemon zest. Use your hands to massage the zest into the sugar for about 20 seconds.

Add the brown sugar and butter to the bowl and use a hand mixer to mix on medium high speed for 2 to 3 minutes. Scrape down the sides of the bowl.

Add the egg and vanilla. Mix again for 20 seconds to combine.

In a separate bowl, whisk together the flour, baking soda, baking powder and salt.

Add the dry ingredients to the butter and egg mixture. Mix on low speed until just combined. Do not overmix.

Add the poppy seeds and mix to combine, again being careful not to overmix.

Use a medium size cookie scoop to divide the dough into 12 equal balls. Place 6 cookie dough balls on each prepared cookie sheet, leaving 2-3 inches between each ball.

Bake the cookies one sheet at a time in the center rack of the oven for 9 to 10 minutes. The edges should be set and the bottoms just beginning to turn a golden brown.

Remove the cookies from the oven and cool on the cookie sheet for 15 minutes before carefully transferring them to a wire rack to cool completely.

To make the glaze, whisk together the sifted powdered sugar and lemon juice or milk until smooth. Adjust consistency as needed, then drizzle over the top of the cooled cookies.