Heat milk, 1/2 of the sugar, and cream in a heavy saucepan until simmering and sugar is dissolved, about 5 minutes, being careful not to scorch milk. Remove from heat and whisk in cocoa powder and dark chocolate until well combined. Allow to cool slightly.

Meanwhile, beat egg yolks and remaining sugar in a large bowl using a whisk or electric beater until thick and pale yellow. Add about 1 cup of the slightly cooled chocolate mixture in a slow stream to avoid cooking eggs, whisking constantly until well combined. Add the remaining chocolate mixture and return the saucepan to medium-low heat. Stir constantly until thickened enough to coat the back of a spoon, about 5 minutes more.

Strain custard through a sieve into a bowl and cover with plastic wrap, touching the surface to prevent skin from forming. Refrigerate for 8 hours to overnight.

Pour cold custard into an ice cream maker and freeze according to manufacturer’s instructions, about 20 minutes. Transfer to an airtight container and freeze until hardened, about 1 hour.