Using an electric hand mixer or stand mixer fitted with the paddle attachment, beat the softened butter on medium speed until smooth and creamy, about 1–2 minutes.
Reduce the speed to low and gradually add the sifted powdered sugar and cocoa powder, about 1/2 cup at a time. Beat after each addition until fully incorporated, scraping down the sides of the bowl as needed.
Add the salt, heavy cream, and vanilla extract. Beat on medium speed until the frosting is light, smooth, and well combined, about 1–2 minutes.
Fold in the melted and cooled chocolate chips using a spatula. Mix gently just until the frosting is evenly blended and glossy.
Check the consistency. If the frosting is too thick, add more heavy cream, 1 tablespoon at a time. If it's too thin, add more powdered sugar. Stir after each addition just until smooth and spreadable.
Use right away or cover and refrigerate.