Place the butter in a medium saucepan or skillet. Cook over medium heat until the butter melts and begins to foam.

Continue cooking, stirring occasionally, until brown specks begin to form at the bottom of the pan and the butter has a nutty aroma.

Transfer the butter (including the browned bits) to a large mixing bowl. Set aside until the butter has solidified (about an hour).

Whisk together the flour, baking powder, and salt. Set aside.

Add the sugar to the cooled brown butter. Using an electric mixer on medium speed, beat the butter and sugar until light and fluffy. Add the eggs, one at a time, mixing well after each addition. Mix in the sour cream and vanilla bean paste.

Reduce mixer speed to low. Gradually add the flour mixture, mixing just until combined.

Transfer the batter to the prepared pan, and spread evenly.

Bake 50 to 60 minutes, or until the cake is golden brown and a pick inserted into the center comes out with moist crumbs. If the top of the cake is getting too brown, cover it loosely with aluminum foil.

Cool the cake on a wire rack for 20 minutes. Then transfer the cake from the pan to a wire rack to cool completely.

Repeat the browning process with the 3 tablespoons of butter.** Place the butter (including the browned bits) in a medium bowl and allow to cool and solidify a bit (about an hour).

Add the cream cheese to the cooled brown butter. Using an electric mixer on medium speed, beat the butter and cream cheese until fluffy. Add the confectioners' sugar, and mix thoroughly.

Add 2 tablespoons of milk, and mix well. Add more milk, a small amount at a time, until the glaze is thick but pourable.

Top the cooled cake with the glaze and allow to set before serving.