Cook butter in small saucepan over medium heat, stirring until light brown and nutty, 3 to 4 minutes. Transfer to bowl and let cool slightly. Refrigerate until solid, about 1 hr. Let stand at room temperature to slightly soften.
Using stand mixer fitted with paddle attachment, add cold brown butter to bowl and beat on medium until light and smooth, scraping bottom of bowl, 3 to 4 minutes.
Add 1 cup of icing sugar and beat until smooth, 2 to 3 minutes, scraping bottom of bowl. Beat in remaining 2 cups icing sugar, cream, vanilla and salt until very smooth. Gradually add cream cheese a few cubes at a time, beating on medium until smooth and light, 3 minutes. Icing makes about 4 ½ cups.
Place 1 cake layer on cake stand or plate. Spread about 1 cup over top, top with another cake layer and repeat, top with remaining cake layer, ¾ cup icing.
Using large offset palette knife, spread remaining icing all over cake and create loose swirly design.