In a food processor, combine chocolate cookies, semisweet chocolate pieces, brown sugar, and cinnamon. Cover and process until finely ground. Add melted butter; cover and process until combined. Press mixture onto the bottom and 2-1/2 inches up the side of an ungreased 10x3-inch springform pan. Set aside.
In a medium saucepan, bring whipping cream just to boiling over medium-high heat. Remove from heat. Add bittersweet chocolate; do not stir. Let stand for 5 minutes. Whisk until smooth. Whisk in coffee liqueur. Pour 2 cups of the ganache into crust, spreading evenly. Cover and freeze for 30 minutes. Transfer the remaining ganache to a small bowl; cover and chill until ready to serve.
Preheat oven to 350 degrees F. In a very large bowl, combine cream cheese, the 1-1/3 cups granulated sugar, and the flour. Beat with an electric mixer on medium to high speed until smooth. In a small bowl, combine coffee powder, rum, the 1 tablespoon vanilla, and the molasses; stir to dissolve coffee powder. Add to cream cheese mixture; beat until combined. Stir in eggs. Pour over ganache layer in pan, spreading evenly.
Place cheesecake on a foil-lined large baking sheet. Bake for 60 to 70 minutes or until a 3-inch area around the outside edge appears set and the center appears nearly set when gently shaken.
Meanwhile, in a medium bowl, stir together sour cream, the 1/3 cup granulated sugar, and the 2 teaspoons vanilla. Pour over hot cheesecake, spreading to cover filling. Bake about 10 minutes more or until topping is set. Cool cheesecake in pan on a wire rack for 15 minutes. Using a small sharp knife, loosen the crust from side of pan. Cool completely (about 1-3/4 hours). Remove side of pan. Cover and chill overnight.
Let cheesecake stand at room temperature for 30 minutes before serving. Heat the remaining ganache. Spoon ganache over each serving of cheesecake. Makes 12 to 16 servings.