1. Preheat oven to 300 degrees. Place strawberries in a single layer on a rimmed baking sheet. Drizzle with corn syrup, and toss gently to coat. Bake until syrup thickens and strawberries turn deep red and shrink slightly, about 1 hour 30 minutes. Transfer strawberries and syrup to a medium bowl, and mash with a potato masher. Let cool completely.

2. Raise oven temperature to 350 degrees. Stir together graham cracker crumbs, 3 tablespoons sugar, and the melted butter in a small bowl. Press mixture into the bottom of a 9-inch springform pan to make an even layer. Bake until crust is firm to the touch and has just darkened, about 10 minutes. Transfer pan to a wire rack, and let crust cool completely.

3. Reduce oven temperature to 325 degrees. Put cream cheese into the bowl of an electric mixer fitted with the paddle attachment; mix on medium-low speed until creamy, about 2 minutes. Scrape down sides of bowl, then gradually add remaining cup sugar and the salt. Scrape down sides of bowl; add eggs, 1 at a time, mixing well after each addition. Scrape down sides of bowl, and mix in vanilla bean seeds and mascarpone until very creamy and no lumps remain, about 3 minutes. (Reserve vanilla bean for another use.)

4. Transfer 5 cups cream cheese mixture to bowl with mashed strawberries; stir to combine. Pour strawberry-cream cheese mixture on top of crust; smooth with an offset spatula. Carefully spoon dollops of plain cream cheese mixture on top, smoothing with an offset spatula.

5. Wrap the exterior of the springform pan in 2 layers of foil; set in a large roasting pan. Fill roasting pan with boiling water until water reaches halfway up the sides of the springform pan. Bake cheesecake until set, about 1 hour to 1 hour 10 minutes. Remove springform pan from water bath, and transfer to a wire rack to cool. Refrigerate until cold, at least 4 hours (up to overnight).