Whisk together sugar, cocoa, and remaining 1 1/2 cups milk in a small heavy saucepan and bring to a boil over moderate heat, stirring. Stir cornstarch mixture again and whisk into chocolate. Return to a boil, whisking, then remove from heat.
Add mint, a pinch of salt, and half of chocolate to saucepan, stirring.
Set saucepan in a large bowl of ice water to cool, whisking frequently.
Freeze chocolate mixture in ice-cream maker. When nearly frozen, add remaining chocolate and churn until blended.