Brown the butter my melting in a sauce pan over medium heat until it foams. Swirl the butter and watch it until it starts to form brown bits in the bottom of the pan. Stop when it is golden and don't allow it to burn.
Set aside to cool fully. When cook, whisk and add the sugar, milk and vanilla. In another bowl whisk the flour, baking powder and salt together. Add the flour to the butter and gently mix to combine.
It will resemble sand and the dough is meant to be fairly dry and crumbly. Gently work it until it starts to come together into a mass. Pour onto a clean counter top and push the dough together until it forms a solid mass. Form into two logs (I did mine about 2-3 inches in diameter although the original recipe said 1 inch). Wrap in plastic wrap and put in the fridge overnight.
Heat oven to 350. Unwrap the dough and slice into rounds about 1/4 inch thick. Roll edges in sugar and place on parchment lined baking sheets. Bake for 12-15 minutes until starting to turn golden on the edges, but barely. Remove from the oven and cool completely.