Preheat the oven to 350°F (175°C). Line a 9 x 5-inch loaf pan with parchment paper.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large mixing bowl, beat the softened butter and granulated sugar with an electric mixer until light, creamy, and well combined.
Add the eggs, lemon zest, vanilla extract, lemon juice, and buttermilk. Mix until combined. The mixture may appear slightly curdled, which is normal.
Gradually add the dry ingredients to the wet ingredients. Mix just until the batter is smooth and no streaks of flour remain. Do not overmix.
Transfer the batter to the prepared loaf pan and smooth the top with a spatula.
Bake for 45-50 minutes, or until the top is lightly golden and a toothpick inserted into the center comes out clean or with a few moist crumbs.
While the cake is baking, stir together the lemon juice and powdered sugar for the syrup until the sugar dissolves.
Allow the baked cake to cool in the pan for 10 minutes. Carefully lift it from the pan using the parchment paper.
While the cake is still warm, slowly brush the lemon syrup over the top and sides. Transfer it to a wire rack and let it cool completely.
To prepare the glaze, whisk the powdered sugar, lemon juice, and milk together until smooth.
Pour or spread the lemon glaze over the completely cooled cake. Let the glaze set briefly before slicing and serving.