1. Preheat your oven to 350°F (175°C) and grease a 9-inch springform pan.

Prepare the Carrot Cake Layer:

1. In a large mixing bowl, combine the grated carrots, sugar, brown sugar, oil, pineapple, egg, and vanilla. Mix well.
2. In a separate bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Gradually mix the dry ingredients into the wet ingredients until combined. Set aside.

Prepare the Cheesecake Layer:

1. In a large bowl, beat the cream cheese and sugar until smooth. Add eggs one at a time, beating well after each addition. Mix in the vanilla extract.
2. Assemble the cake by pouring half the carrot cake batter into the prepared pan. Spoon the cheesecake batter over it evenly, then top with the remaining carrot cake batter.
3. Bake for 50-60 minutes or until the center is set. Allow the cake to cool completely before removing it from the pan.

Prepare the Frosting:

1. Beat the cream cheese and butter together until smooth. Gradually add powdered sugar, then vanilla extract. If the frosting is too thick, add milk a teaspoon at a time to reach the desired consistency.
2. Frost the cooled cake and refrigerate for at least 4 hours before serving.