Preheat the oven to 325 degrees F. Place a rack on the lower third shelf in the oven. Wrap the 2-piece cheesecake pan with 2 layers of heavy-duty aluminum foil. Set aside.
Pour 1 of the 12-ounce packages of thawed cherries into a colander or sieve that is set over a small bowl. Let the liquid drain off while you prepare the recipe. Very gently (so that you do not crush the cherries) press to drain more liquid.
Melt the chocolate in a medium bowl over a double boiler, stirring until melted, or in the microwave in 20-second increments until warm. Stir to melt. Set aside.
Add the cookies and sugar to a food processor bowl fitted with the chopping blade. Pulse until coarse crumbs.
Add the melted butter and pulse again until fine crumbs and the butter has moistened all the crumbs. (Stop and stir as needed while processing the cookies.)
Pour the crumbs into the 9” cheesecake pan. Using the flat bottom of a glass, the back of a spoon, or a silicone spatula, press the crumbs onto the bottom and about 1 ½ to 2” up the sides of the pan.
Bake the crust for 10 minutes.
Remove the crust from the oven to a cooling rack to cool while you make the filling.
Place the cream cheese in the bowl of a stand mixer fitted with the paddle attachment. Stir the cream cheese on medium-low until smooth, approximately 1 to 2 minutes, scraping the bowl as needed. (Alternatively you can make the batter with a hand-held mixer.)
Slowly add the sugar and vanilla.
Turn the mixer to low and add the eggs and yolk one at a time, mixing well after each addition.
Maintaining low speed, add the sour cream. Stop and scrape the mixer’s bowl and beater if necessary. Combine on low until all the ingredients are smooth.
Pour approximately half of the batter into the bowl with the melted chocolate. (Your chocolate should not be completely cooled, but still a little warm.) Stir with a silicone spatula to thoroughly combine. Pour the chocolate batter into the crust, carefully smoothing the top and the chocolate batter to the edge.
Scatter the (12 ounces) thawed, drained cherries on top of the chocolate batter. Reserve the liquid for the topping.
Gently pour the remaining batter evenly over the cherries. Smooth the top.
Place the foil-wrapped cheesecake pan into the larger pan/roasting pan and place it on the rack in the oven. Pour in enough water to come about 2” up the sides of the pan. Be sure not to get any water into the foil or onto the cheesecake. (Pushing the cheesecake to one side of the larger pan helps to avoid splashing water. Then carefully scoot the pan back to the center.)
Bake the cheesecake for 1 hour 10 to 1 hour and 25 minutes, or until the edges are slightly puffy, yet the middle wobbles like jello. You may also check the internal temperature of the cheesecake with a probe thermometer. It should read at least 145 degrees. The residual heat will allow the cheesecake to continue to rise in temperature, which is fine.
Turn the oven off. Use a heat-safe spatula to barely prop the oven door open. Leave the cheesecake in the oven for 1 more hour.
Remove the cheesecake from the oven and from the water bath. Unwrap the foil and discard. Allow the cheesecake to come to room temperature, another 2 to 4 hours.
Loosely cover the cheesecake and refrigerate overnight, 8 hours is a good minimum.
Between the time of the cheesecake baking and a few hours before serving, make the cherry topping. This part of the dessert can be made up to 2 days in advance. Add the remaining 12-ounce package of cherries and all their liquid to a small saucepan. Add the reserved liquid from the drained cherries you placed inside the cheesecake. Bring the cherries up to a gentle simmer over medium heat. Add sugar to taste. Add the cherry balsamic vinegar or lemon juice if using.
Stir to dissolve the sugar.
Combine the cornstarch and water in a small bowl. Add to the cherries and stir until slightly thickened. Your cherries should still be simmering.
Taste and adjust sugar and lemon juice/balsamic as desired. Cool slightly before covering well and refrigerate until completely cooled. (You can allow the cherries to cool to room temperature if you didn’t make them in advance to refrigerate.)
To serve, slice the cheesecake and place slices onto individual plates. Top with cherry topping. For pretty, clean slices, run the knife under hot water, wipe dry and slice again.