Butter the bottom of a loaf pan (9x5) and cover with parchment paper, leaving enough parchment hanging over all sides.
Whisk together all the dry ingredients in a bowl including ¼ teaspoon baking soda, 2 ½ cups bread flour, and ⅛ teaspoon kosher salt. Keep aside.
Beat together 1 cup butter, zest from 3 lemons, and 1 ¾ cup granulated sugar in a small bowl of a stand mixer fitted with a paddle attachment for 8 minutes on medium-high, scraping down the sides in between.
Add in ⅓ cup sour cream and continue beating for another minute.
Start adding eggs and egg yolk, one by one, scraping down the bowl after each addition.
Add in ¼ cup lemon juice and continue beating for a minute, scraping down the sides and bottom before adding flour.
Set your stand mixture on the lowest setting and add in the flour mix. Stir until flour is incorporated, for a minute. Scrape the bottom and sides of the bowl by using a spatula and finish folding the ingredients together. The batter will be velvety and thick.
Transfer batter to a loaf pan and smooth top by using a spatula.
Bake at 325 F until skewer inserted comes out clean from the center, for about 1 hour and 35 minutes.
Meanwhile, prepare the lemon soaking syrup. Boil all the sauce ingredients in a saucepan until no longer cloudy. Once done, let it cool.
Once the cake is baked, take it out from the oven and let it cool for another 10 minutes.
Pour over half of the lemon syrup and allow to soak in. Then pour in the remaining syrup and leave for 15 minutes.
Now run a sharp knife along the edges and lift the cake out by using the parchment sling and set on a baking rack.
While it cools, make the icing. Melt butter in a medium microwave-safe bowl for 10 to 20 seconds. Add the confectioners' sugar and whisk well while adding lemon juice.
Pour icing over the cake evenly and spread it. Slice and serve!