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To Make the Chocolate Cookie Crust:
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Preheat the oven to 300 degrees F. Lightly spray a 9-inch springform pan with nonstick cooking spray.
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Line the pan with parchment paper and lightly spray the parchment and sides of the pan.
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In a food processor, grind the cookies to a very fine crumb.
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You should have about 3 1/2 cups.
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Put the crumbs in a small bowl.
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Pour the melted butter over them and mix with a rubber spatula until well combined.
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Turn the crumb mixture into the prepared pan and press it into the bottom and up the sides, leaving about 1/2 inch between the top of the crust and the top of the pan.
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Use the back of a large spoon to get an even layer of crust.
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Place the pan in the freezer and let the crust set for about 10 minutes.
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Bake the crust in the oven until it is dry to the touch, about 10 minutes.
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Transfer the pan to a wire rack and let cool.
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To Make the Flourless Chocolate Cake:
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Increase the oven temperature to 350 degrees F.
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Using a double boiler or microwave, melt the butter and chocolate together.
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Set aside to cool.
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In a small bowl, whisk together the espresso powder, coffee, salt, and vanilla.
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Set aside.
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In the bowl of a standing mixer fitted with the whisk attachment, beat the egg yolks with 1/2 cup of the sugar until the mixture is light and has almost doubled in volume, about 5 minutes.
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Add the chocolate mixture and beat until just combined.
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Scrape down the sides and bottom of the bowl and mix on low speed for 5 seconds.
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Add the coffee mixture and beat until just combined.
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Scrape down the sides and bottom of the bowl and mix on low for 5 seconds.
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In a clean bowl fitted with the whisk attachment (or you can elect to do this step by hand if you are feeling strong), beat the egg whites until foamy.
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Gradually increase the speed to high and add the remaining 1/2 cup sugar, beating until soft peaks form.
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Scoop 1 cup of the egg whites into the chocolate mixture.
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Use a rubber spatula to gently fold in the egg whites.
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After about 30 seconds of folding, add the remaining egg whites and continue folding until they are almost completely combined.
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Do not rush the folding process, work gently, and take care not to overmix.
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Pour the batter onto the cooled cookie crust and bake for 38 to 42 minutes, until the cake is set but still jiggles slightly.
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It might not appear to be completely cooked.
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Transfer it to a wire rack and cool completely.
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(As it cools, the cake will deflate in the center and look sunken.
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Do not despair, this is just the way it settles.)
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Tightly wrap and refrigerate the cake for at least three hours or overnight.
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To Make the Chocolate Pudding:
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In a medium saucepan, whisk together the sugar, cocoa powder, cornstarch, and salt.
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Add the egg yolks and whisk until combined.
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The mixture will look like a thick paste.
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Slowly pour in the milk, whisking constantly.
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In a saucepan over medium heat, bring the mixture to a boil, whisking constantly to prevent it from burning on the bottom of the pan.
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Boil for 30 seconds, then transfer it to a medium bowl.
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Add the butter, vanilla, and chocolate and whisk until combined.
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Continue to whisk for a few more minutes to cool the mixture slightly.
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Let the pudding stand for 15 minutes at room temperature.
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Press a piece of plastic wrap directly onto the surface of the pudding to prevent a skin from forming, and chill it for at least 3 hours.
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To Assemble the Mississippi Mud Pie:
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Stir the pudding to loosen it, then pour it on top of the cake, making sure to stay inside the cookie-crust border.
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Use an offset spatula to spread the pudding into an even layer.
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Return the cake to the refrigerator for at least 30 minutes while you prepare the whipped cream topping.
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Spread whipped cream across the pudding layer, all the way out to the sides, unmold the cake, and serve it immediately.
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The cake can be kept, covered, in the refrigerator for up to 2 days.
