Combine blackberries and sugar in a pot over the stove. Cook over medium heat for about 10 minutes, stirring often. The berries should break down and the mixture should bubble and thicken up. Use a wooden spoon to mash the berries up while cooking. Mix together corn starch and cold water into a slurry and add to the blackberry mixture. Continue to cook over medium heat for 1-2 more minutes while mixture thickens. Remove from heat and stir in vanilla extract. Place in an airtight container and store in the refrigerator until ready to use.
Graham cracker crust
Preheat oven to 350°. Prepare an 8X8 or 9X9 pan by greasing and lining with parchment paper.
Using either a food processor or a rolling pin and a plastic storage bag, crush the graham crackers into a fine crumb. Add the melted butter, salt, sugar and cinnamon. If using the food processor pulse together until it is combined and resembles wet sand. Otherwise combine all of the ingredients in a bowl and stir to combine.
Press mixture into prepared baking pan and bake in preheated oven on the middle oven rack for 10 minutes. Set aside to cool while you make the cheesecake. Turn oven down to 325°.
Cheesecake
In the bowl of a stand mixer using the paddle attachment or using a hand mixer, combine the room temperature cream cheese and granulated sugar on medium speed until fully combined, about 2-3 minutes. Scrape down the bottom and sides of the bowl routinely. Add in each egg one at a time and combine after each addition. Add in the heavy whipping cream, vanilla bean paste or extract, lemon juice, and zest. Mix on medium speed until combined, about 30 seconds. Turn speed up to medium-high and mix for another minute.
Pour cheesecake batter into prepared graham cracker crust. Drop spoonfuls of the blackberry jam onto the batter. Using a butter knife swirl gently. Bake on the middle oven rack at 325° for 30-35 minutes. The cheesecake will still jiggle slightly in the center but will appear set when it is finished. Set on a wire rack to cool at room temperature for one hour. After one hour tightly wrap with plastic wrap or foil and store in the refrigerator for 6-8 hours, or overnight. Once completely cooled, slice and serve. Top with fresh blackberries if desired.
Notes
Cheesecake bars will remain fresh in the refrigerator for 3-4 days. They can also be frozen in an airtight container for about a month.