Place heavy cream, milk, and 1/2 cup plus 2 tablespoons sugar in a medium saucepan and bring to a boil over medium-high heat. Remove from heat; set aside.
Prepare an ice-water bath and set aside. In a medium bowl, whisk together egg yolks and remaining 1/2 cup plus 2 tablespoons sugar until well combined. Whisking constantly, slowly add 1/3 of the hot cream mixture to the egg yolk mixture. Slowly whisk egg yolk mixture into saucepan with remaining cream mixture until well combined. Add chocolate and continue whisking until chocolate is melted and well combined.
Strain mixture through a fine mesh sieve into a medium bowl set over ice-water bath. When cool, transfer mixture to a refrigerator until chilled, at least 8 hours and up to overnight.
Transfer mixture to an ice cream maker and prepare according to manufacturers' instructions. Keep ice cream frozen until ready to use.