Start by browning enough butter to account for water evaporation. You'll need 300g of brown butter, so start with at least 340g.
To brown the butter, in a small saucepan, heat the butter over medium heat, stirring frequently. The butter will be ready once the fat solids have separated and darkened, and the butter is an even medium amber color.
Transfer to a bowl and place in the fridge to solidify again, stirring every 10 minutes or so. Alternatively, place the bowl over a larger bowl of ice water, and whisk constantly until it becomes solid, yet soft.
Once the brown butter is solid, yet soft, stir to mix in all of the browned bits that have settled at the bottom.
In the bowl of a stand mixer fitted with a paddle attachment, beat the cream cheese and brown butter on high speed until smooth.
Slowly add the powdered sugar and beat until fully combined. Continue beating on low speed until no more lumps remain and the frosting is smooth.