Preheat the oven to 350°F. Line an 8 1/2 x 4 1/2 inch loaf pan with parchment paper.
Whisk the eggs, milk and vanilla together in a large bowl.
In another large bowl, whisk together cake flour, sugar, lemon zest, baking powder and salt.
Add the butter and half of the egg mixture to the dry ingredients.
Beat on low speed until incorporated. Increase the speed to high and beat for 1 minute. Scrape down the sides of the bowl.
Add the remaining egg mixture in two parts. Beat well for 20 seconds and scrape down the sides of the bowl after each addition.
Pour the batter into the prepared loaf pan and spread it evenly to fill the corners. Bake for 55 to 60 minutes in a loaf pan, until a toothpick inserted into the loaf comes out clean.
Meanwhile, prepare the lemon simple syrup. Warm the sugar and lemon juice on the stove on low, stirring until the sugar has dissolved. Remove from heat and set aside.
Remove the loaf from the oven and place it on a wire rack to cool. Brush the top with half of the syrup. Let cool in the pan for 10 minutes.
Run a knife around the edges of the loaf to loosen it from the pan. Use the parchment paper to remove the loaf from the loaf pan. Peel off the parchment.
Poke the sides and bottom of the loaf with a skewer or toothpick. Brush the rest of the syrup on the bottom and all four sides.
Place right-side-up on a cooling rack and let cool completely. Wrap with plastic wrap and store for 24 hours.
Make the icing. Combine powdered sugar, lemon juice, milk, butter, vanilla and salt in a bowl and whisk or beat together until smooth and somewhat thick. Add more powdered sugar 1/4 cup at a time if you desire a thicker icing.
Drizzle the icing over top of the cake. Allow to harden, approximately 1-2 hours. Slice and serve.
Notes
Make the syrup while the cake bakes to save some time.
Don't skip the syrup — it keeps the cake moist while it rests overnight.
You need to plan ahead — make this 1 day ahead of when you want to serve it.