Place the butter in a small saucepan and brown very well over a low-medium heat. If you haven’t browned butter before, see the notes section for some handy links.
Once browned, pour the butter into a silicon cake tin and place in the fridge or freezer to set. It will take about 15-20 minutes in the freezer.
Once the butter is completely cooled, break it up and place it in the bowl of your stand mixer. Beat the butter with the whisk attachment until it is light and fluffy, scraping down the sides as necessary. Don’t be fooled – this isn’t a quick process. It will take about 10-15 minutes of beating and scraping down. The mixture should be light and fluffy – not quite white but a very pale beige.
While this is happening, sieve the icing sugar and the cocoa together. I hate sieving so I wouldn’t instruct you to do this unless it was totally necessary for a smooth icing, I promise.
When the butter is ready, turn the speed to low and mix in the icing sugar mixture and the milk. It might form a cloud of icing if you go too quickly. Add the salt and vanilla bean paste (stop the machine to add vanilla or it gets tangled in the beaters) and mix until combined. Taste and adjust if necessary.