In a 2 to 3 quart pan over high heat, bring milk and mint just to a boil. Remove from heat, cover, and let stand 20 minutes.

Meanwhile, in a bowl, mix sugar and cornstarch. Whisk in egg yolks.

Pour milk mixture through a fine strainer into another bowl; discard mint. Whisk about 1/4 of the mint-milk into the egg yolk mixture. Add cocoa and whisk until blended; pour back into pan. Add remaining milk to pan.

Return pan to medium heat and stir with a whisk until mixture begings to bubble, 2 to 4 minutes.

Remove cocoa mixture from heat and pour into a bowl Stir in yogurt. Cover surface of mixture with plastic wrap and chill until cold, at least 2 hours or up to 1 day.

Pour cold mixture into an ice cream maker and freeze according to manufacturer's directions. Midway through freezing, add chopped chocolate; continue freezing until ice milk is firm enough to mound. Serve, or cover and freeze up to 3 days. If making ahead, let the dessert soften in refrigerator about 30 minutes before serving.