Preheat oven to 350°F (180°C). Grease a 9x5-inch (or 8x4-inch) loaf pan and line it with a strip of parchment paper leaving an overhang on two sides to lift out the cake.
In a small bowl combine the flour, baking powder, baking soda, lemon zest, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment (or using an electric mixer) cream the softened butter and sugar together at medium-high speed until pale and fluffy, about 4 minutes. Scrape the sides of the bowl as needed.
With the mixer running on low speed, add the eggs one at a time, then beat in the vanilla extract and lemon juice. Beat on medium-high speed until combined.
With the mixer on low, add about one-third of the flour mixture and mix until almost combined, then add half the buttermilk and mix until just combined. Repeat with another third of flour mixture and then the last half of the buttermilk, ending with the last third of the flour. Beat until just incorporated. Add poppy seeds and fold in with a spatula.
Scrape the batter into the prepared pan and bake for 45-50 minutes until the cake is golden brown and a toothpick comes out mostly clean with only a couple moist crumbs. Baking times vary, so keep an eye on yours.
Let the cake cool for about 10 minutes in the pan. Stir together the lemon juice and confectioners’ sugar for the lemon syrup. Carefully transfer the cake to a cooling rack, then brush the syrup all over the cake while it’s still warm. Allow cake to cool completely.
When the cake is cooled, make the icing. Start with adding 1 Tbsp lemon juice to the powdered sugar then add as much milk as needed. The icing should be thick and not too runny. Spoon icing over cake and let harden before serving.