Brown butter over medium heat in small saucepan until golden & nutty smelling.
Cream hot brown butter and cream cheese together in stand mixer with whisk attachment until fluffy and smooth, 5-10 minutes.
Add 1 cup powdered sugar followed by 1-2 tablespoons milk until incorporated. Mix in vanilla and milk until icing is smooth and spreadable.
Spread over warm king cake or cinnamon rolls, or eat by the spoonful with a glass of wine.
Store chilled in airtight container up to five days.