Melt the butter in a saucepan set over medium heat. The butter will get foamy and bubble. Cook until the butter solids in the bottom of the pan begin to turn deep golden brown. Immediately remove from heat to prevent burning.
19 tablespoons unsalted butter
Pour the browned butter through a fine-mesh sieve into a bowl. Allow to cool and refrigerate until cold but still soft. Reserve 2 tablespoons of brown butter for the glaze. Set aside.
Sift together the flour, baking powder, and salt.
2 cups all-purpose flour, 1 teaspoon baking powder, ¼ teaspoon kosher salt
Preheat oven to 350ºF. Grease a Bundt cake pan, dust with flour, and shake off excess. Set aside.
In a large bowl, using a hand mixer, beat the brown butter and sugar until light and fluffy. Add the eggs, one at a time, and mix to combine.
1½ cups granulated sugar, 2 large eggs
Add the dry ingredients in 3 parts, alternating with the buttermilk in 2 parts. Add vanilla.
¾ cup buttermilk, 1 teaspoon pure vanilla extract
Pour the batter into the prepared Bundt 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 together until smooth.
2 tablespoons brown butter, 2 teaspoons milk, ¼ teaspoon vanilla extract, 1 cup powdered sugar
Pour the glaze over the cooled cake. Allow to cool/harden.