Pour the buttermilk into a 2-cup liquid measuring cup, stir in the poppy seeds, and set aside at room temperature for at least 2 hours.
Preheat the oven to 350 degrees F. Thoroughly spray the inside of a Bundt pan with the baking spray and set aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream the butter and 2 cups of the granulated sugar on medium speed for about 5 minutes, until light yellow and fluffy. With the mixer on low, add the eggs, one at a time, the vanilla, and lemon zest, scraping down the bowl with a rubber spatula.
Sift the flour, cornstarch, salt, baking powder, and baking soda into a medium bowl. Add 1/4 cup of the lemon juice to the buttermilk mixture. With the mixer on low speed, alternately add the flour mixture and buttermilk mixture in thirds, beginning and ending with the flour. Scrape the bowl with a rubber spatula to be sure the batter is well mixed. Spoon the batter into the prepared pan, smooth the top, and bake for 40 to 50 minutes, until a cake tester comes out clean.
Meanwhile, place the remaining 1/2 cup of granulated sugar and the remaining 1/2 cup of lemon juice in a small saucepan and cook over high heat until the sugar dissolves. Set aside. When the cake is done, allow it to cool in the pan for 10 minutes, then turn it out onto a baking rack set over a large plate. Spoon the warm lemon syrup slowly over the cake, allowing it to be absorbed into the cake. Set aside for at least 30 minutes to cool.
For the glaze, whisk the confectioners' sugar and lemon juice together in a small bowl, adding a little more sugar or lemon juice to make a smooth, thick, but pourable glaze. Drizzle over the cake, allowing it to drip down the sides. Transfer to a flat cake plate and serve at room temperature.