In a large measuring cup, whisk together eggs, egg yolk, vanilla, and buttermilk. Set aside.
In the bowl of an electric mixer, whisk together flour, sugar, baking powder, and salt. Mix the dry ingredients on low speed. Cut the butter into small pieces and drop into the running mixer. Run until the butter is worked in and looks like it has been incorporated.
Slowly pour the buttermilk mixture into the flour. When all of the liquid has been added, raise the speed to medium and beat until the batter is light and fluffy, about 2 minutes.
Bake the cupcakes in a muffin tin lined with paper liners in a preheated 350 degrees for 20 to 35 minutes. Let cool in the pan for 10 minutes, then cook completely on a wire rack.
To make the icing, put brown sugar, butter, and salt in a medium saucepan. Melt over medium heat. Bring the mixture to a boil and add milk and vanilla. Boil for 3 minutes, stirring occasionally.
Add the powdered sugar all at once and beat with a wooden spoon or spatula until the icing is thick and smooth.
You do not have a lot of time with this icing, so work quickly. Dip the tops of the cupcakes into the icing. Roll them around a little to smooth out the icing and make sure you cover the whole top. Let dry on a rack.