Preheat your oven to 350°F (175°C). Grease and flour a bundt pan.
In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In another bowl, beat together the oil, granulated sugar, and brown sugar until well combined. Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Fold in the grated carrots.
In a separate bowl, beat the cream cheese, granulated sugar, egg, and vanilla extract until smooth.
Pour half of the carrot cake batter into the prepared bundt pan. Spoon the cheesecake mixture over the batter, then top with the remaining carrot cake batter.
Bake for 60 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.