Brown the butter: Melt the butter in a skillet or small saucepan over medium heat. Swirl or stir the butter with a wooden spoon as it starts to foam and sputter. When the butter begins to turn golden brown and smells nutty, about 1 minute. Remove from heat, strain through a fine sieve or coffee filter, and let cool to room temperature.
Meanwhile, place the confectioners' sugar, 1/4 cup whole milk, and corn syrup in the bowl of an electric mixer fitted with a whisk. Beat to combine. With the mixer on medium, drizzle in the brown butter a little at a time, waiting until it is absorbed before adding more. Beat in another teaspoon of milk to make the frosting creamier, if necessary. Use on cakes, cupcakes, or cookies.