In a medium pot over medium heat, combine raspberries, granulated sugar, cornstarch, and lemon juice. Bring to a boil, stirring often, and cook until raspberries are completely broken down, 6 to 8 minutes.

Strain sauce through a fine-mesh strainer into a medium bowl; discard solids. Let cool completely.

Make Ahead: Sauce can be made 3 days ahead. Transfer to an airtight container and refrigerate.

Preheat oven to 325°. Wrap outside of an 8" or 9" springform pan in foil. In a food processor or resealable bag, crush cookies into fine crumbs. Transfer crumbs to a medium bowl. Add granulated sugar and salt and stir to combine. Add butter and stir until mixture holds together when pressed with your fingertips and resembles wet sand.

Press cookie mixture into bottom and up sides of prepared pan.

Bake crust until lightly golden, 12 to 15 minutes. Let cool.

In the large bowl of a stand mixer fitted with the paddle attachment, beat cream cheese and granulated sugar until smooth. Add eggs, one at a time, beating well after each addition. Add yogurt, lemon zest and juice, vanilla, and salt and beat until combined.

Pour about half of batter over cooled crust. Spread a thin layer of raspberry sauce over top. Gently pour remaining batter over sauce. Dot sauce all over top. Using a toothpick or knife, swirl sauce into batter. Reserve remaining raspberry sauce for serving.

Place springform pan in a large roasting pan. Place roasting pan in oven. Carefully pour in enough boiling water to come about 2" up the sides of springform pan.

Bake cheesecake until center only slightly jiggles, 60 to 75 minutes.

Turn off oven, prop door open, and let cheesecake cool in oven 1 hour.

Remove foil from outside of pan. Refrigerate until chilled, at least 4 hours or up to overnight.

Top cheesecake with raspberries. Serve remaining raspberry sauce alongside.