In a large measuring cup, whisk together, the eggs, egg yolk, vanilla, and buttermilk and set aside.
In the bowl of an electric mixer, whisk together the flour, sugar, baking powder and salt. Fit the mixer with the paddle attachment and turn the mixer on low. With the mixer running, gradually drop in the butter pieces and mix until the texture is uniform and the bits of butter aren’t discernable, about 2 minutes. With the mixer still on low, begin slowly pouring in the wet ingredients. When all the wet ingredients have been added, crank the speed up to medium and mix until the batter is light and fluffy, about 1 1/2 minutes.
Portion the batter into the muffin tins and bake until a toothpick comes out clean and the tops spring back when lightly touched, 20-25 minutes. Let cool in the pans for about 10 minutes, then transfer them to a a wire rack to cool completely.
Put the dark brown sugar, butter and salt in a medium saucepan, and melt them together over medium heat, stirring often. Bring the mixture to a boil and add the milk and vanilla. Boil for 3 minutes, stirring occasionally. Dump in the confectioners’ sugar all at once and beat with a wooden spoon until the icing is thickened and smooth. Quickly ice the cupcakes by holding each by the base and dipping the tops in the hot icing, rolling them slightly to coat evenly. Turn them quickly upright so the icing will even out while it’s still warm and will set smoothly. Store any leftovers in an airtight container for up to 4 days.