In a medium bowl, beat the butter at high speed until creamy (1 to 2 minutes).
Add the cocoa powder and vanilla; continue beating, scraping the bowl often, until the mixture is creamy (1 to 2 minutes).
Add the powdered sugar, 1 cup at a time, beating well after each addition. Add the milk and ¼ cup sour cream; continue beating, scraping bowl often, until creamy (1 to 2 minutes).