Cook and stir condensed milk and cocoa in a medium saucepan over low heat until smooth and slightly thickened, about 5 minutes. Remove from heat and allow to cool slightly. Stir in heavy cream, light cream, and vanilla. Refrigerate until cold, about 1 hour.
Pour mixture into the canister of an ice cream maker and freeze according to manufacturer's directions. Stir in marshmallows and nuts halfway through the freezing process.
