Preheat oven to 350 degrees. Spray a 13x9-inch pan with baking spray.
Using an electric mixer, beat butter, sugar, salt, baking powder, and vanilla until light and fluffy, at least 5 minutes.
Add the eggs and yolks one at a time, beating well after each addition.
Mix in buttermilk just until incorporated.
Add flour and mix just until incorporated. Scrape down the sides of the bowl to make sure everything is evenly mixed.
Pour batter into prepared pan. Bake for 35 minutes, or until cake springs back when pressed lightly in the middle. Cool.
Melt the butter in a saucepan over low heat. Turn heat off and whisk in brown sugar and milk until the sugar is dissolved.
Stir in powdered sugar until smooth. Slowly pour over cooled cake. You can let it cool slightly if too runny, but don't let it cool too much or it will be difficult to spread on the cake.