In a medium saucepan, whisk together the heavy cream, cocoa powder, sugar, and salt until smooth.
Place over medium heat and bring to a gentle simmer, whisking often, until the sugar and cocoa are fully dissolved (about 3-5 minutes). Do not let it boil hard.
Remove from heat. Add the chopped chocolate and whisk until completely melted and smooth.
Whisk in the whole milk and vanilla extract.
Pour into a heatproof bowl or container, cover, and refrigerate until thoroughly chilled (at least 2 hours, or overnight for best results).
Whisk the chilled base well, then pour into your ice cream maker. Churn according to the manufacturer's instructions (typically 20-30 minutes) until it reaches soft-serve consistency.
Transfer to an airtight container. For firmer ice cream, freeze 2-4 hours (or overnight). Let it sit at room temperature a few minutes before scooping if it's too hard.