Preheat the oven to 350. Grease a 9x5-inch loaf pan and line with a strip of parchment paper, leaving an overhang on two sides to lift out the loaf.
In a small bowl, combine flour, baking soda, baking powder, salt, and lemon zest. Set aside.
In a medium bowl, mix together softened butter and sugar. Use a stand mixer or hand mixer. Mix for about 4 minutes on medium-high speed until the mixture is pale and fluffy. Scraped the sides of the bowl if needed.
With your mixer on low speed, add one egg at a time, then beat in the vanilla extract and lemon juice. Turn the mixer to medium-high speed and mix until the ingredients are well combined.
With your mixer on low speed, add ⅓ of the flour mixture until just combined. Add half the buttermilk and mix until just combined. Add another ⅓ of the flour mixture, and then the other half of the buttermilk. Mix the last third of the flour mixture until just incorporated. Add poppy seeds and fold in with a spatula.
Pour your batter into the prepared loaf pan and bake for 45-50 minutes, until the loaf is golden brown and a toothpick inserted into the center comes out clean. Baking times may vary, so keep an eye on yours.
Let the loaf cool in the pan for about 10 minutes. Stir together the powdered sugar and lemon juice for the lemon syrup. Transfer your loaf to a cooling rack and brush the syrup all over the warm loaf.
Once your loaf is cooled, make the icing. Add 1 tbsp lemon juice to the powdered sugar, then add as much milk as needed. Your icing should be thick and not too runny. Spoon the icing over the top of your loaf and let it harden. Serve and enjoy!