In a metal bowl set over a pan of lightly simmering water, melt chocolate. Gradually whisk in the milk and heat, stirring constantly, until smooth. Remove from the heat and let cool.
In a large bowl, whisk eggs until light and fluffy, 1 to 2 minutes. Whisk in sugar, a little at a time, then continue whisking until completely blended, about 1 minute.
Add the cream, vanilla and salt and whisk to blend. Pour the chocolate mixture into the cream mixture and blend.
Cover with plastic wrap. Chill in freezer until very cold (40 degrees) about 30 to 60 minutes.
Start ice cream maker and add the mixture. Mix until ice cream has the consistency of soft serve ice cream, about 25 to 30 minutes.
Remove from ice cream machine canister and transfer to an airtight container. Freeze until firm, at least 3 hours.