Place the butter in a medium saucepan or skillet over medium heat. Cook, stirring occasionally, until brown specks begin to form at the bottom of the pan and the butter has a nutty aroma.
Add the sugar to the cooled brown butter. Use an electric mixer on medium speed to beat the butter and sugar until light and fluffy, then beat in the eggs one at a time, followed by the sour cream and vanilla bean paste.
Pour the batter into the prepared pan, spreading evenly. Bake 50 to 60 minutes, or until the the cake is golden brown and a pick inserted into the center comes out with moist crumbs.