Preheat your oven to 350°F (175°C) and grease a 9-inch round springform pan.
In a bowl, mix the cream cheese, granulated sugar, and flour until smooth. Add the eggs one at a time, then mix in the vanilla extract and sour cream.
In another bowl, whisk together the oil, crushed pineapple, granulated sugar, brown sugar, eggs, and vanilla extract for the carrot cake mixture.
Add the spices, flour, baking soda, baking powder, and salt to the wet ingredients, mixing until just combined. Fold in the grated carrots and pecans.
Pour 1 1/2 cups of the carrot batter into the prepared springform pan. Add half of the cheesecake mixture, then layer the remaining carrot batter and finish with the rest of the cheesecake mixture.
Bake for 60 to 65 minutes, until the center is slightly jiggly but not liquid. Cover with foil if browning too quickly.
Let the cheesecake cool for an hour, then refrigerate for at least six hours, preferably overnight.
For the frosting, beat together the cream cheese, butter, and vanilla extract. Gradually add the confectioners' sugar until smooth.
Once chilled, frost the cheesecake and top with chopped pecans.
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Notes
Store leftovers in the refrigerator for up to one week.