Add the white chocolate chips and 1/3 cup of milk to a glass, microwave safe bowl. Heat in the microwave at 50% in 30 second intervals, stirring each time, until the the white chocolate is smooth. Allow to cool.
Cream butter until light and fluffy, about 3 minutes. Stir in melted and cooled white chocolate and then gradually add in confectioner's sugar, about 1 cup at a time. Then add in milk, 1 tablespoon at a time, followed by vanilla. Beat on high for about 1 to 3 minutes until the buttercream frosting is smooth, fluffy and will hold a peak.
If you want to make the buttercream frosting ahead, place it into an airtight container and refrigerate. When ready to use, let the frosting come to room temperature, then beat it again with the mixer until fluffy.