Stir together the raspberries, water, sugar, and cornstarch in a saucepan. Bring to a boil and cook until the sauce thickens, about 5 minutes. Strain through a mesh strainer and discard the seeds. Set aside.
Preheat the oven to 325°F (162°C). In a bowl, stir together the cookie crumbs, 3 tablespoons of sugar, and butter. Press the mixture into the bottom of a 9-inch springform pan. (I used Oreo cookies; I placed the cookies in a Ziploc bag and smashed them with the bottom of a small pan.)
Melt the white chocolate chips with half-and-half in a metal bowl set over a pan of simmering water, stirring occasionally until smooth.
Gently beat together the cream cheese and the remaining ½ cup of sugar with an electric mixer on low speed until smooth. Beat in the eggs one at a time. Stir in the cornstarch, vanilla extract, and white chocolate mixture. Mix well, but do not overmix. Pour half of the batter over the crust and drizzle 3 tablespoons of sauce on top. Add the remaining batter and spoon another 3 tablespoons of sauce on top. Use a knife tip or a bamboo skewer to swirl the sauce into the batter to create a marbled effect.
Bake until the center jiggles slightly and the edges are puffed, about 40-45 minutes. Cool at room temperature, then cover with plastic wrap and chill for at least 2 hours, or preferably overnight, before removing from the pan. Top the cheesecake with the remaining sauce and fresh raspberries. Slice and serve cold.