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