Bring 2 1/4 cups milk just to a boil over medium heat in a medium-sized saucepan.

While milk is heating, whisk sugar, 1/4 cup cold milk, cornstarch, and salt together in a mixing bowl until smooth. Whisk cold mixture into the boiling milk and bring to a boil over medium heat; let boil for 3 minutes, whisking constantly until mixture is very thick. Remove from heat.

Place chocolate into a mixing bowl.

Bring remaining 1/2 cup cold milk just to a boil over medium heat in a small saucepan. Pour hot milk over chocolate and let stand until chocolate melts, about 1 minute. Whisk until smooth.

Stir chocolate mixture into cornstarch-milk mixture. Force through a fine-mesh sieve into a mixing bowl. Let cool slightly, stirring frequently to prevent a skin from forming. Cover surface with wax paper. Chill in a refrigerator until cold, at least 1 1/2 hours.

Pour mixture into a gelato or ice cream maker. Freeze according to manufacturers instructions, about 15 minutes. Transfer to an airtight container and freeze in a freezer until hardened if necessary.