In a stand mixer fitted with a paddle attachment, whip the butter on medium speed until creamy and light in color, about 4 minutes. Scrape down the bowl and paddle, then add half of the powdered sugar. Mix on low speed until just incorporated, then add the rest of the powdered sugar. With the mixer still in the off position, sift in the black cocoa powder and add the vanilla, milk, and salt. Mix on low until all ingredients are thoroughly combined, about 2 minutes, scraping down the bowl and paddle as needed.
Place the buttercream in an airtight container and allow the color to deepen for 24 hours or more. You can keep it at room temperature for one day or store it in the refrigerator for up to two weeks. When you're ready to use it, bring it back to room temperature and re-mix with your stand mixer on low speed until the consistency is smooth.
