Preheat the oven to 350°F. Line a 1-pound loaf pan with parchment paper, letting it fold over the sides for easier removal.
To make the batter: In the bowl of a stand mixer fitted with the paddle attachment, add the butter, cane sugar, and vanilla extract. Mix on low until combined and there are no butter chunks.
Add in the eggs and mix on low until combined. Scrape down the side of the bowl and mix again until combined.
Add the milk, cake flour, sea salt, and lemon zest in that order, and mix on low until combined, scraping down the sides of the bowl as needed to ensure a smooth batter and everything is mixed.
Pour the cake batter into the loaf pan and spread evenly.
Bake for 1 hour 15 minutes or until a wooden pick inserted in center comes out clean. Allow to cool completely in the cake pan.
To make the glaze: In a medium bowl, add the powdered sugar and lemon juice. Whisk until combined into a smooth glaze.
Spread the glaze over the top of the loaf cake.
Store in an airtight container for up to 3 days.