Make the sauce: Combine the raspberries and sugar in a medium saucepan over medium heat. Cook, stirring frequently, until the mixture comes to a simmer. Turn the heat to medium-high and continue to cook, while stirring, until it is thickened, 4 to 5 minutes. Pour the mixture into a fine mesh sieve set over a bowl, and press the raspberry purée through with a spatula, scraping the underside of the sieve. Discard the remaining seeds. You should have about ¾ cup/180 milliliters of raspberry purée. Chill the purée while you prepare the rest of the cheesecake.
Make the crust: Pre-heat oven to 325. Add the graham crackers to the bowl of a food processor fitted with the blade attachment and pulse until you have fine crumbs. Add the butter, sugar and salt, and pulse until the butter is evenly distributed and the crumbs are moistened. Spray a 9-inch springform pan* with cooking spray, and pour the crumb mixture into the pan. Use a measuring cup or flat-bottomed glass to very firmly press the mixture into the bottom and about 1½ inches up the sides of the pan. Bake in hot oven for 10 minutes. Move to refrigerator and allow to cool completely.
Make the filling: In the bowl of a stand mixer fitted with the paddle attachment, combine the cream cheese, ricotta cheese, sugar, lemon juice, vanilla extract and salt. Mix on low speed until combined, then turn the mixer up to medium and beat until smooth and fluffy, about 2 minutes. Move to a separate bowl.
Make the filling: Clean the bowl of the mixer and switch to the whisk attachment. Add the heavy cream, turn up to high and whip until thick and fluffy, about 5 minutes. Stop the mixer once or twice to scrape the bottom and sides of the bowl. Fold the whipped cream into the cheese filling.
Assemble the cake: Pour the mixture into the prepared crust and use an offset spatula to smooth the top. Dot the top with ¼ cup/60 milliliters of chilled raspberry purée, about 1 teaspoon per dollop. Using a paring knife or offset spatula (for thick swirls) or wooden skewer (for fine swirls), swirl the purée into the cream cheese mixture. Chill at least 8 hours and up to overnight.
*I didn't have a springform pan so I just used an 8-inch cake pan. I fitted the pan with parchment paper that came up the sides so I could lift the cheese cake out when it was done. I still sprayed the pan/parchment paper with cooking spray before adding in crust. Since it calls for a 9" pan and I used an 8", I saved about a 1/3 of a cup of the crust crumb mixture and used it to sprinkle on top when serving.
