Preheat the oven to 350°F (175°C). In a mixing bowl, combine graham cracker crumbs, sugar, and melted butter. Mix until well combined.

Press the mixture into the bottom of a springform pan. Bake for 10 minutes, then remove and let cool.

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

In another bowl, beat the eggs and mix in oil and grated carrots. Combine wet and dry ingredients until just mixed. Fold in walnuts if using.

Pour the carrot cake batter over the cooled crust and spread evenly.

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

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

Bake in the preheated oven for 50-60 minutes, or until the center is set and slightly jiggles. Turn off the oven and leave the cheesecake inside for 1 hour.

Remove from the oven and let cool to room temperature, then refrigerate for at least 4 hours or overnight.

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

Spread the frosting over the chilled cheesecake before serving.