1. Heat the oven to 325°F. Grease a 9-inch springform pan, line the bottom with parchment paper, and wrap the outside with foil if using a water bath.

2. Prepare the brownie batter with the brownie mix, eggs, oil, and water or brewed coffee according to the package directions.

3. Spread the brownie batter into the prepared pan and bake for 18 to 22 minutes, until the top looks set but the center is still slightly soft. Cool for 10 minutes.

4. Beat the softened cream cheese and sugar until smooth. Add sour cream and vanilla, then mix until creamy.

5. Add the eggs one at a time on low speed, mixing only until combined. Mix in the flour or cornstarch.

6. Pour the cheesecake filling over the partially baked brownie base and smooth the top.

7. Place the pan in a larger roasting pan and add hot water around it until it reaches about 1 inch up the side of the springform pan.

8. Bake for 55 to 70 minutes, until the edges are set and the center still has a gentle jiggle.

9. Turn off the oven, crack the door, and let the cheesecake cool inside for 45 to 60 minutes.

10. Remove from the oven, cool completely, then refrigerate for at least 6 hours or overnight.

11. Warm the hot fudge sauce until pourable, then spoon it over the chilled cheesecake.

12. Top with whipped cream, chopped nuts or crunchy toppings, brownie crumbs if using, and maraschino cherries before slicing.