Preheat oven to 350°F (175°C). Grease a 9-inch springform pan.
In a bowl, combine flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In another bowl, whisk together oil, granulated sugar, brown sugar, eggs, and vanilla extract.
Mix dry ingredients into wet ingredients until just combined. Fold in grated carrots.
Pour carrot cake batter into the prepared pan and spread evenly.
In a large bowl, beat cream cheese and sugar until smooth. Add eggs one at a time, beating well after each addition. Stir in vanilla extract.
Pour cheesecake batter over carrot cake layer in the pan.
Bake in preheated oven for 50-60 minutes, or until the center is set. Let cool completely before removing from pan.