Preheat oven to 325F. Line two or more cookie sheets with parchment paper cut to fit.
Melt both chocolates (but not the chips) and the butter in a double boiler over simmering water. When melted, cool slightly.
Sift the flour, baking powder, and salt together into a small bowl and set aside.
Using an electric mixer on medium speed, beat the eggs, vanilla, and espresso powder in a medium sized mixing bowl until they are mixed together, about 20 seconds. Add the sugar to the egg mixture and blend until thick, about 1 minute. Scrape the bowl. Add the melted chocolate/butter mixture and blend 1 minute more. Scrape the bowl. Add the flour on low speed and mix until blended, about 20 seconds. Add the chocolate chips and stir in by hand or on low speed.
Drop the dough (it’s very gooey) by large tablespoons full about 2 inches apart onto the parchment-lined cookie sheets. Bake until the cookies rise slightly and form a think crackly crust, 11-13 minutes. Let cool for a minute or two then remove to more parchment paper or a rack to cool completely.
In a medium saucepan over medium low heat combine the milk and heavy cream. Split the vanilla bean and scrape the seeds into the milk mixture and then add the empty pod to it, too. Simmer it all over low heat for 30 minutes, then remove and discard the bean pod.
In a mixing bowl beat together the sugar, cocoa, espresso powder, eggs and egg yolk until thick. Measure about 1 cup of the hot milk/cream mixture and, with the mixer on low, add in a thin stream to the egg/sugar mixture, mixing thoroughly. Stir the chocolate chips to the hot milk/cream that remains in the pan and once the chips have mostly melted, add the egg/sugar mixture to it. Cook over low hear stirring constantly until it thickens and looks like chocolate pudding. Remove to a heatproof bowl, cool ever so slightly and then cover with plastic wrap pressed to the surface (so that a skin does not form on the top). refrigerate until completely chilled, 4 to 6 hours or overnight.
Spoon the chocolate mixture into the ice cream maker (it won’t pour if it’s thoroughly chilled) and mix until thickened, about 30 minutes. It may be served immediately or further firmed in the freezer until ready to serve. Serve with sweetened whipped cream.