This frosting will take a little time because the melted browned butter must come back to room temperature/cool after cooking it. You could make the browned butter earlier in the day, or while the cakes are cooking and cooling.
First, add 3 sticks of the unsalted butter to a small saucepan over medium-high heat.
Using a rubber spatula or wooden spoon, stir continuously until butter starts to bubble and turn golden brown. You want to watch it very carefully so it doesn't burn. The flecks on the bottom the pan should turn brown and the liquid butter should turn an amber color. It should smell nutty!
Once the browned butter is a nice ambery color, with brown flecks on the bottom, remove from heat. Pour in a heat proof bowl or container. Place in the fridge to cool, until it starts to solidify. Or you can leave on the counter until it comes to room temperature.
Once solidified, it's ready to make the browned butter frosting. In a stand mixer with the paddle attachment (or in a large bowl with a hand mixer), add the browned butter and beat on a high speed until it's completely creamy. If the butter chilled all the way this will take longer, until the hardened pieces incorporate. Keep beating it!
Next add the softened cream cheese and continue beating, for 5-6 minutes, on high speed, until very fluffy and light.
Next add the vanilla, then alternate adding powdered sugar and a tablespoon of milk, until you've added all the powdered sugar in. Frosting should be light and fluffy. Add tiny bits of milk or cream to reach a desired consistency.
Once completed, frost onto completely cooled cakes. Enjoy!