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Combine 2 1/2 cups blueberries and 1 tablespoon cornstarch in a blender. Blend until smooth.
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Cook puree in a small saucepan over medium-heat for 13 minutes, stirring constantly.
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Remove from heat and cool to room temperature. Reserve 1/3 cup cooked puree for topping.
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Preheat oven to 325°F. Grease a 9-inch springform pan.
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Beat cream cheese in a large bowl with an electric mixer on medium speed until creamy.
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Whisk together 1 cup sugar, remaining 2 tablespoons cornstarch, and salt in a small bowl. Gradually add to cream cheese, mixing well.
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Add eggs, beating well after each addition (beat on medium speed).
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Stir in 1 teaspoon vanilla.
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Pour batter into prepared springform pan.
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Dollop blueberry puree on batter. Gently swirl puree into batter with a knife.
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Bake for 1 hour, or until set. Remove from oven and cool on wire rack for 20 minutes. The cheesecake will crack. It's okay because it will be covered up.
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Stir together sour cream, 2 tablespoons sugar, and 1/2 teaspoon vanilla in a small bowl. Spread on top of cheesecake.
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Bake for an additional 10 minutes. Cool cheesecake on a wire rack to room temperature. Cover and chill overnight.