In a medium skillet over medium heat, melt butter. Reduce heat to medium-low and continue to cook, stirring often and watching closely, until butter simmers and foams, then starts to turn golden brown, 5 to 7 minutes.

Immediately pour butter into a large heatproof bowl, scraping browned bits from bottom of skillet into bowl. Let cool to room temperature, then refrigerate until butter is thickened but not solid, 10 to 20 minutes.

Arrange a rack in center of oven; preheat to 350°. Grease a 9" x 5" loaf pan with cooking spray.

In a medium bowl, whisk flour, baking powder, and salt.

To bowl with brown butter, add brown sugar and granulated sugar. Using a handheld mixer on medium-high speed, beat until light and fluffy. Add eggs, one at a time, beating to blend after each addition. Beat in vanilla. Add dry ingredients and beat on medium-low speed until a just few dry streaks remain. Beat in milk until just combined. Pour batter into prepared pan; smooth top.

Bake cake until a tester inserted into the center comes out clean, about 1 hour. Transfer to a wire rack and let cool.

In a medium skillet over medium heat, melt butter. Reduce heat to medium-low and continue to cook, stirring often and watching closely, until butter simmers and foams, then starts to turn golden brown, 5 to 7 minutes.

Immediately pour butter into a large heatproof bowl, scraping browned bits from bottom of skillet into bowl. Let cool slightly.

Whisk in confectioners’ sugar and salt until smooth. Add cream and vanilla; whisk until combined. Spread frosting over cake.