Preheat your oven to 350°F and prepare a 9x13" cake pan with non-stick spray and parchment paper.

Bring your 5 eggs to room temperature by placing them in a bowl of warm water for around 10 minutes.

In a large bowl, whisk together the flour, sugar, baking powder, baking soda, and salt.

Add the softened butter to the dry ingredients and beat until the mixture resembles sand and there are no visible chunks of butter, about 1-2 minutes.

Bring the milk to room temperature. Stir in the white vinegar.

In a medium bowl, lightly whisk together the eggs, egg yolks, vanilla extract, and milk mixture.

Add ½ of the liquid to the dry ingredients and beat on medium speed for 1 ½ minutes, scraping down the sides of the bowl as needed.

Add ½ of the remaining liquid and beat until combined. Finally, add the remaining liquid and beat until just combined.

Pour the batter into your prepared cake pan. Bake at 350°F for 45-50 minutes, or until a toothpick inserted into the middle comes out with a few crumbs.

Allow the cake to cool completely in the pan.

In a medium pot, add the melted butter and brown sugar. Bring to a boil over medium-high heat and boil for two minutes.

Whisk in the milk and then bring just to a boil. Remove from the heat and allow the mixture to cool to lukewarm.

Add the powdered sugar and beat until fully combined.

Immediately pour the icing over the cooled cake, using an offset spatula as needed to push the icing over the surface of the cake.

Allow the icing to set for 20-30 minutes before slicing and serving.

Store leftover cake in an airtight container at room temperature.