Heat milk, sugar, cocoa powder, espresso powder, and salt in a saucepan over medium heat, stirring often, until milk is barely simmering.
Beat egg yolks in a bowl; gradually stir in about 1/2 cup hot milk mixture. Beat until smooth. Transfer egg yolk mixture back to the saucepan and continue cooking and stirring until thickened, about 5 minutes.
Meanwhile, melt dark chocolate in a microwave-safe bowl in 30-second intervals, stirring after each melting, for 1 to 3 minutes.
Remove saucepan from heat and stir melted chocolate into milk-egg mixture; transfer mixture to a large bowl, cover, and refrigerate until well-chilled, 4 hours to overnight.
Stir cream and vanilla extract into chilled chocolate mixture and pour into an ice cream maker. Freeze according to manufacturer's directions until it reaches "soft-serve" consistency. Transfer ice cream to a lidded container; cover surface with plastic wrap and seal. For best results, ice cream should be placed in the freezer at least 2 hours or overnight.
I use a Lindt® Special Dark Chocolate bar in this recipe. Dark chocolate chips are nearly as good. I have an in-pan mixer, so I mix this custard furiously at every stage (taste it right along; you may want to increase sugar slightly or adjust coffee up or down). I freeze the mixture in my Donvier® ice cream freezer, 1-quart size. Try it! It will knock your socks off.