Pour the mixture into a clean glass or stainless-steel bowl and lay plastic wrap directly on the surface to prevent a skin from forming on top. Allow the mixture to sit 30 to 45 minutes, until no longer hot. Then place in the refrigerator to cool completely, about 4 hours. If the mixture needs to be used right away, submerge most of the bowl in an ice bath and let it sit 30 to 40 minutes, refreshing the ice as necessary.
Once the mixture has cooled completely and thickened further, pour it into the bowl of the gelato machine, gently stir in the dark chocolate and ladyfingers, and churn the gelato according to the manufacturer's directions. The gelato will expand and should spin until it's thick and creamy but still soft enough to scoop into a storage container, about 30 to 55 minutes.
Using a rubber spatula, scoop the gelato into a storage container.
Press a piece of plastic wrap or parchment paper directly on the surface of the gelato, seal the container with an airtight lid, and put it in the freezer.
Freeze at least 4 to 5 hours. When ready, the gelato should be firm enough to scoop but soft and creamy in texture.