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Preheat oven to 325°F (163°C) and grease a 9-inch springform pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Set aside.
In a large bowl, combine the granulated sugar, brown sugar, eggs, oil, and vanilla. Mix well.
Stir in the grated carrots and optional nuts. Gradually fold in the dry ingredients until fully incorporated.
Pour the carrot cake batter into the prepared pan, spreading it evenly.
In a separate bowl, beat the cream cheese, granulated sugar, eggs, vanilla, and sour cream until smooth. Add the flour and mix until smooth.
Pour the cheesecake mixture over the carrot cake layer, smoothing the top.
Bake for 45-55 minutes, until the center is set and a toothpick in the cheesecake comes out clean.
Cool in the pan on a wire rack at least 1 hour, then refrigerate at least 4 hours or overnight to set.
For the frosting: beat the powdered sugar and vanilla with the remaining cream cheese until smooth. Spread over the chilled cheesecake before serving.