Pre-heat oven to 375°F. Prepare a donut pan (or 2 if you have more than 1) with butter and flour and set it aside.
In a large bowl, cream the butter and sugar until light and fluffy (start on low speed and then increase to medium speed) with an electric mixer.
Add the egg and vanilla extract and mix at low speed until combined.
Mix the espresso powder with the boiled water until dissolved. Add the mixture to the batter and mix until smooth and creamy (it should not look curdled) about 1 to 2 minutes.
In a medium bowl whisk together the flour, baking powder, baking soda, and salt.
Alternate adding the dry ingredients to the batter with the buttermilk, beginning and ending with the flour mixture.
Mix on medium-low speed until combined.
Add 3 heaping tablespoons to each prepared donut well. Smooth batter with a spoon and tap donut pan on the counter to remove air bubbles.
Bake the donuts for 10 minutes until the donuts are golden. Allow donuts to cool for 5 minutes in the pan. After 5 minutes transfer to a cooling rack to fully cool.
Prepare the coffee glaze while the donuts are cooling. Mix the half and half with the espresso powder until dissolved and set aside.
In a medium deep bowl, whisk together powdered sugar with the espresso/cream mixture until it is a smooth and silky texture and streams from the whisk.
Dip the tops of the donuts in the glaze and return to a cooling rack with parchment paper or a cookie sheet underneath to catch any dripping glaze. If using sprinkles add them before icing sets.