In a medium-size bowl, stir butter until smooth. Add sugar and stir until well combined (mixture will be thick). Add the egg and egg yolk and vanilla. Stir until well combined and smooth.
Sprinkle the baking soda and salt over the top of the butter mixture and stir to combine. Add ½ cup of the flour. Stir just until the flour is incorporated. Add approximately half of the buttermilk and stir until combined.
Repeat adding another ½ cup of the flour and then the remaining buttermilk, stirring until blended after each addition. Add the final ½ cup flour and stir until the flour has disappeared.
Pour batter into the prepared pan and bake for 35-45 minutes or until a toothpick inserted in the center comes out clean. (Cover the top with foil during the last 5-10 minutes if getting too brown.)
Remove from oven and allow to cool for 10 minutes in the pan then invert onto a wire rack to cool completely.
Add the half and half (or cream), the vanilla bean paste, 2¼ cups of the powdered sugar and the salt. Stir with a whisk until smooth and creamy. If the icing is too thin, add a bit more powdered sugar (up to another ¼ cup). If it's too thick, thin it with a little bit more half and half.
When the cake is completely cooled scrape the icing from the bowl onto the center of the cake. Spread over the top (and down the sides, if desired) with an offset spatula. You want a thick, but spreadable icing.
See Café Tips above in post for lots of additional tips and instructions.
Cake recipe adapted from Food.com
Candied Walnut recipe adapted from Preppy Kitchen.