Melt butter in a saucepan over medium heat. Butter will get foamy and bubble. Cook until butter solids in the bottom of the pan begin to turn deep golden brown. Immediately remove from heat.
Pour through a sieve into a bowl. Allow to cool and refrigerate until cold but still soft. Reserve 2 tablespoons of brown butter for the glaze.
Sift together flour, baking powder, and salt.
Preheat oven to 350ºF. Grease a Bundt cake pan, dust with flour shaking off excess.
In a large bowl, beat butter and sugar until light and fluffy. Add eggs, one at a time, and mix to combine.
Add the dry ingredients in 3 parts, alternating with the milk in 2 parts. Add vanilla.
Pour batter into the pan.
Bake for 45-50 minutes or until a toothpick inserted into the middle sections comes out clean.
When the cake is done, transfer it to a cooling rack. Remove the cake from the pan. Place a plate over the top of the cake and flip the cake over. Let cool completely.
For the glaze, in a medium bowl, beat brown butter, powdered sugar, vanilla, and milk until smooth.
Pour the glaze over the cooled cake