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Preheat oven to 350 degrees.
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In medium bowl, beat butter until fluffy. Add powdered sugar and mix well. Add eggs one at a time beating well after each. Beat in flour on low until mixture forms small crumbs. Add half and half. Press evenly into a 9 x 13 baking pan. Bake 20 minutes or until lightly browned.
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In a large bowl, beat cream cheese with the 3/4 cup sugar until smooth. Add eggs one at a time. Mix in vanilla. Pour over the shortbread crust.
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In a medium bowl, stir chopped apples together with 3 tablespoons sugar and cinnamon. Spoon over cheesecake mixture.
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To make the walnut streusel topping, combine all ingredients in a medium bowl. Use a pastry blender or fork to mix. It should form small crumbs.
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Sprinkle topping over the apples. Bake 40 - 45 minutes until filling is set.
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Before serving, drizzle with caramel topping.