Step 1

In a medium light-colored skillet, melt 16 tablespoons (2 sticks) of the butter over medium-high heat. Cook, swirling the pan occasionally, until the milk solids are golden brown and the butter smells nutty, 4 to 6 minutes. Immediately transfer the butter to a large bowl (this will be for the cake batter); be sure to scrape out the browned bits from the bottom of the skillet. Wipe the skillet out and repeat the same process with the remaining 3 tablespoons butter. Transfer the butter to a medium bowl (this will be for the glaze). Set both bowls aside until the butter has solidified to the texture of softened butter.

Step 2

Preheat the oven to 350 degrees F. Butter a 4 1/2- by 8 1/2-inch loaf pan.

Step 3

To prepare the cake, whisk together the flour, baking powder and salt in a medium bowl and set aside. Add the granulated sugar to the 2 sticks of browned butter and beat the mixture with an electric mixer on medium-high speed until light and fluffy, 3 to 4 minutes. Add the eggs one at a time, beating after each addition. Then add the vanilla bean seeds. Beat in the flour mixture just until combined. Transfer the batter to the prepared pan.

Step 4

Bake until the cake is golden brown and a toothpick inserted in the center comes out with moist crumbs attached, 60 to 70 minutes. Transfer the cake to a rack to cool for 20 minutes, then remove the cake from the pan and let cool completely.

Step 5

Add the cream cheese to the 3 tablespoons browned butter and beat with an electric mixer on medium until fluffy. Beat in the confectioners' sugar. Add enough milk to the mixture to make a thick but pourable glaze. Top the cooled cake with the glaze before slicing.