In a medium bowl, combine the crushed Oreo cookies with melted butter and 1/2 teaspoon of espresso powder. Mix until the crumbs are evenly coated. Press the mixture firmly into the bottom of a 9-inch springform pan. Chill in the refrigerator while you prepare the filling.
In a heatproof bowl, combine the bittersweet chocolate and 1/3 cup of heavy cream. Melt gently over a double boiler or in the microwave, stirring until smooth. Set aside to cool slightly.
In a large mixing bowl, beat the cream cheese, granulated sugar, light brown sugar, Dutch-process cocoa powder, and 1 and 1/2 teaspoons of vanilla extract until smooth and creamy. Gradually add the melted chocolate mixture, mixing until fully incorporated.
In a separate bowl, whip the remaining 1 cup of heavy cream until stiff peaks form. Gently fold the whipped cream into the chocolate cream cheese mixture until fully combined. This will give your cheesecake a light and airy texture.
Pour the cheesecake filling over the prepared crust and spread it evenly. Smooth the top with a spatula. Cover the cheesecake with plastic wrap and refrigerate for at least 4 hours or overnight for best results.
Once the cheesecake has set, prepare the ganache by heating 1 and 1/4 cups of heavy cream and 1/4 cup of butter in a saucepan over medium heat. Stir until the butter is melted and the mixture is hot but not boiling. Remove from heat and add the finely chopped semi-sweet chocolate and 1 teaspoon of espresso powder. Let it sit for a minute, then stir until smooth.
Pour the chocolate ganache over the chilled cheesecake, allowing it to drip down the sides. Chill again for about 30 minutes to set the ganache.
Before serving, top the cheesecake with optional garnishes like espresso beans or chocolate shavings. Slice and enjoy the no-bake espresso chocolate cheesecake with friends and family!