Preheat the oven to 350°F (175°C). In a mixing bowl, combine graham cracker crumbs, sugar, and melted butter. Press the mixture into the bottom of a 9-inch springform pan.

In a large bowl, whisk together flour, granulated sugar, brown sugar, baking powder, baking soda, cinnamon, nutmeg, and salt.

In another bowl, mix together vegetable oil, eggs, grated carrots, and walnuts (if using). Combine wet and dry ingredients until just mixed.

Pour the carrot cake batter over the crust in the springform pan and spread evenly.

In a mixing bowl, beat the softened cream cheese until smooth. Add sour cream, sugar, and vanilla extract, mixing until combined.

Add eggs one at a time, mixing on low speed until just blended. Pour the cheesecake batter over the carrot cake layer.

Bake in the preheated oven for 60 minutes or until the center is set. Allow to cool at room temperature, then refrigerate for at least 4 hours or overnight.

In a mixing bowl, beat together softened cream cheese and butter until creamy. Gradually add powdered sugar and vanilla extract, mixing until smooth.

Once chilled, remove the cheesecake from the springform pan and frost the top with the cream cheese frosting. Slice and serve.