Brown your butter in a saucepan over medium-low heat. Start by melting the butter. Then, your butter will start to foam and froth. Next, the foaming will stop, and you may see white flecks of solidified milk fat floating around in your pan. Finally, the foam and forth will come back a little. Move your foam and froth aside so you can see into the pan. When you see dark brown flecks, remove the pan from the heat and let it cool.
After your browned butter has cooled, put it in a heat-safe dish. Place that dish in the fridge to cool completely. You want your butter to be solid again before making frosting with it.
You can wait to mix your frosting until your cupcakes are filled if you wish to. This is what I do!
To mix the frosting, just place the chilled browned butter, powdered sugar, maple syrup and 2 tablespoons of milk or heavy cream in a large mixing bowl. Mix until it forms a smooth frosting mixture.
If your frosting seems too thick, add more milk or heavy cream by the tablespoon until you reach your desired consistency.