Bring milk to a simmer. Meanwhile, combine egg yolks and sugar in the bowl of an electric mixer. Beat at medium-high speed until very thick and pale yellow, 3 to 5 minutes.
Add half the milk to egg-yolk mixture and whisk until blended. Stir into remaining milk and cook over low heat, stirring constantly, until mixture is thick enough to coat a spoon.
Remove from heat and immediately stir in cream. Pass mixture through a strainer into a medium mixing bowl set in an ice bath. Whisk in melted chocolate and let chill, then freeze in an ice-cream maker according to manufacturer's instructions. Store in a plastic container in the freezer.