In the top of a double boiler or microwave, melt the chocolate chips, stirring until the mixture is smooth. Set aside to cool.
In the bowl of a stand mixer using the paddle attachment, beat the butter on high until light and fluffy, about 5 minutes.
With the mixer on low, gradually add the powdered sugar, scraping down the sides of the bowl.
Stir the melted white chocolate before adding to the butter mixture. Beat on medium speed for 2 minutes until the mixture is smooth. Add vanilla and gradually add milk until you get a smooth yet stiff consistency.
If the buttercream is not stiff, refrigerate about 10 minutes before piping. This just means the butter got too warm.
Notes
Be sure to read the post for tips and tricks to make this buttercream.