To start making this delicious cake, preheat your oven to 350 degrees F (175 degrees C) and prepare a 9-inch cake pan by greasing and flouring it. In a separate bowl, combine the flour, cocoa powder, baking soda, and salt and whisk together. Set the dry mixture aside. In another bowl, mix together the milk, food coloring, vinegar, and 1/2 teaspoon of vanilla extract.
In a large bowl, use an electric mixer to beat the softened butter and 3/4 cup sugar until the mixture is light and fluffy. The mixture should become visibly lighter in color. Add the egg and beat until smooth. Gradually add in the flour mixture and milk, alternating between the two until everything is just incorporated. Pour the batter into the prepared pan.
Bake the cake in the preheated oven for about 30 minutes or until a toothpick inserted into the center comes out clean. Let it cool in the pan for 10 minutes before removing and cooling it completely on a wire rack.
After the cake has cooled, cut half of it into cubes and set them aside. Cut the remaining cake into 1/2-inch strips and place them onto a baking sheet. Put the cake strips back in the oven and bake until they have dried out, which should take about 15 minutes. Flip the strips over halfway through cooking to dry them evenly. After they're completely dry, let them cool to room temperature and then crush them into fine crumbs.
Mix the cake crumbs with melted butter until they're evenly moistened. Press the mixture into a 10-inch springform pan and refrigerate it until the butter has hardened, which should take about 45 minutes.
While the crust is hardening, whip the whipping cream until it's stiff and set it aside. In a separate bowl, beat the softened cream cheese with 2/3 cup sugar and 2 teaspoons of vanilla extract until the mixture is smooth. Fold in the whipped cream until it's evenly mixed. Pour half of the cream cheese mixture into the hardened crumb crust, then spread the cake cubes evenly over the cream cheese mixture. Finally, spread the remaining cream cheese over the top. Refrigerate the cheesecake for at least 4 hours until the cream cheese has set and the cheesecake is firm.
