Combine cream, milk, and salt in a medium heavy-bottomed saucepan. Cook over medium-high heat until simmering. Add mint leaves. Remove from heat, cover, and let steep at least 30 minutes, until the cream has a clear mint flavor.
When mixture is ready, whisk yolks and sugar together in a medium bowl (don’t do this ahead, as the sugar will “set” the yolks). Drizzle 1/2 cup cream mixture into eggs, whisking constantly. Whisk in rest of cream, 1/2 cup at a time. Pour mixture back into saucepan.
Cook over medium heat, stirring constantly with a wooden spoon, until mixture thickens slightly and coats the spoon, 3 to 4 minutes; be careful not to overcook, or you’ll scramble the yolks.
Pour custard through a fine-mesh strainer into a clean bowl; discard mint. Set bowl in a larger bowl of ice and cold water and let cool completely. Remove from ice bath and chill, covered, at least 2 hours and up to overnight.
Chop chocolate, put in a glass bowl, and microwave in 15-second bursts until halfway melted. Remove and stir until completely melted. Line a baking sheet with parchment paper and spread chocolate evenly over parchment. Put in freezer. Once cold, break chocolate into quarter-size pieces and freeze until ready to use.
Freeze custard in an ice cream maker according to manufacturer’s instructions. In the last minute of churning, add the chocolate and chopped mint. Eat right away or, for a firmer ice cream, freeze at least 4 hours.