Step 1

Heat chocolate and milk in a medium heatproof bowl set over a saucepan of barely simmering water, stirring occasionally, until smooth. Set aside to cool.

Step 2

Using an electric mixer on medium-low speed, beat cream cheese, butter, and salt in a large bowl until smooth. Add powdered sugar and beat on low speed until incorporated, then increase speed to medium and beat until light and fluffy, about 2 minutes. Scrape down sides of bowl and add chocolate mixture. Beat just until smooth, then beat in vanilla.

Step 3

Chill frosting, stirring occasionally, until it holds soft peaks, 15–20 minutes for milk chocolate and 50–60 for white chocolate. Beat briefly with electric mixer on low speed until smooth, then spread evenly over cupcakes.