In a medium saucepan, add in the blackberries, sugar, lemon juice, and lemon zest. Cook over medium heat for about 20 minutes or until the blackberries are completely broken down.
Strain out the seeds in the blackberry sauce with a fine mesh strainer.
Place back in the saucepan over medium heat.
In a small bowl, add in the cornstarch and about 1/4 cup of the strained blackberry sauce. Whisk until combined and there are no lumps.
Add the cornstarch mixture into the saucepan with the rest of the blackberry sauce and whisk until combined.
Let the sauce come up to a boil while stirring constantly. Once the sauce comes up to a boil, let it boil for 1 minute to thicken the sauce. Remove from heat and let cool.
Preheat oven to 325 degrees. Spray the sides of a 9-inch springform pan with nonstick baking spray. Wrap the sides of the pan with a strip of parchment paper.
Mix the crushed Oreos, melted butter, and sugar until it resembles wet sand and can be packed down.
Press the crust evenly into the bottom of the prepared pan.
Place in the oven for 10 minutes. Remove from the oven and let cool while you work on the cheesecake filling.
Add the cream cheese to a large mixing bowl. Beat with a hand mixer on medium until the cream cheese is smooth. Scrape down the sides of the bowl.
Add in the sugar and beat on medium again until combined.
Add in the eggs one at a time and beat in each egg completely before adding the next. Scrape down the bowl after adding each egg.
Add in the sour cream, heavy cream, lemon, and lemon zest. Beat on medium until the mixture comes together. It will be runny.
Pour the batter on top of the crust in the prepared pan.
Pour the blackberry sauce evenly on the cheesecake. Using a skewer or knife, swirl the sauce into the cheesecake.
Wrap the springform pan with 2 layers of heavy-duty tin foil. Place the cheesecake into a slightly bigger pan and fill with one inch of water.
Place in the oven with a tented piece of tin foil over the top of the cheesecake to prevent browning on the top. Let the cheesecake bake for 1 hour 25 minutes or until done. To make sure the cheesecake is done, gently give it a wiggle. If it slightly jiggles in JUST the middle like jello, then it's done. If it jiggles all over, it needs to bake longer. Turn off the heat and crack the oven door slightly. Let it cool in the oven for 1 to 2 hours until it reaches room temperature and then place in the fridge to chill overnight.
Remove the cheesecake from the springform pan. Slice and serve.