Combine the milk and shiitake mushrooms in a saucepan and bring to a boil over medium heat, stirring frequently. Remove from heat, cover plastic wrap and steep mixture for at least 4 hours.
Taste the milk mixture to ensure it has absorbed a good, strong shiitake flavor. Then re-heat mixture and pour it through a fine-meshed strainer. Discard the mushrooms. Re-measure the milk and adjust it to equal 3/4 cup.
Combine cream with mushroom-infused milk in a saucepan and bring to a boil over medium heat.
While cream is heating, put the sugar and water in a medium heavy-bottomed pot. Use an unlined copper pan if you have one. Stir to mix the water and sugar. Place over medium heat and cook, stirring occasionally with a wooden spoon, until the sugar turns dark amber, 4 to 5 minutes. To check the color, dab a small amount of the syrup on a white plate. If any crystals form on the sides of the pan as the sugar darkens, wash them down with a wet pastry brush.
When the sugar is the correct shade, remove the pan from the heat and put a sieve or spatter guard over it. Wearing an oven mitt, slowly pour the hot cream into the sugar syrup a little at a time. The mixture will splutter and foam. Be careful, as it is very hot. When the mixture stops bubbling, whisk it to incorporate any caramel stuck to the bottom.
Place the egg yolks in a medium bowl, and whisk by hand until blended. Whisk about 1/2 cup of the caramel mixture into the yolks to warm them gradually. Whisk in another 1 cup, and then whisk in the rest. Add the chocolate and whisk until it melts.
Strain the custard through a fine-mesh sieve into a bowl.spoon the custard into twelve 2 1/2 ounce espresso cups (or small, 3 ounce ramekins) filling them three-fourths full. Let the custard cool to room temperature.
Preheat the oven to 300F. Bring a large teakettle of water to a simmer.
Put the cups in a large baking pan. Pour the hot water into the baking pan to reach halfway up the sides of the cups. Cover the pan with aluminum foil.
Bake on the middle shelf of the oven until the tops are set but the entire custard jiggles when a cup is moved, about 25 minutes. Immediately remove the cups from the hot water. Let cool to room temperature.
Cover the cup and refrigerate for at least 6 hours or up to overnight. Bring to room temperature before serving.