Place butter in a small saucepan over medium heat. Cook until butter turns slightly brown (small brown specks will appear on bottom of saucepan). Stir continuously. Be careful not to overcook (if you see black specks, you have overcooked). Remove from heat, cover, and allow to cool until solidified (you can place in refrigerator for 20-25 minutes to speed up this process).
In the bowl of an electric mixer fitted with the paddle attachment, beat butter for 2 minutes on medium speed. Adjust speed to low and add powdered sugar to butter 1 cup at a time until well incorporated. Add vanilla extract, and heavy cream (or milk). Beat on medium for 3 minutes. Add more powdered sugar to thicken frosting or milk/cream to loosen frosting.
Browned butter has a strong, nutty flavor that many people love! Try it on a ginger or spice cake!