Preheat oven to 350°F (175°C). Wrap the outside of a 9-inch springform pan with a double layer of heavy-duty aluminum foil, ensuring it's watertight. This is for the water bath later.

In a medium bowl, combine the graham cracker crumbs, chopped pecans, brown sugar, and cinnamon. Pour in the melted butter and stir until all the crumbs are evenly moistened.

Press the crumb mixture firmly and evenly into the bottom of the prepared springform pan. Bake for 10 minutes. Set aside to cool while you prepare the cake layer.

In a medium bowl, whisk together the flour, cinnamon, baking soda, baking powder, ginger, nutmeg, and salt. Set aside.

In a separate large bowl, whisk the granulated sugar, brown sugar, and vegetable oil together until well combined. Whisk in the egg and vanilla extract.

Gradually add the dry ingredients to the wet ingredients, mixing on low speed (or by hand) until just combined. Do not overmix. Fold in the finely grated carrots.

Pour the carrot cake batter evenly over the pre-baked crust. Bake at 350°F (175°C) for 20-25 minutes, or until a toothpick inserted into the center comes out with moist crumbs attached. Let the pan cool on a wire rack for about 15 minutes. Reduce oven temperature to 325°F (163°C).

In the bowl of a stand mixer fitted with the paddle attachment (or using a hand mixer), beat the softened cream cheese on medium-low speed for 2-3 minutes until completely smooth and free of lumps. Scrape down the sides and bottom of the bowl frequently.

Add the granulated sugar and flour, and beat on low speed until just combined. Scrape the bowl again.

Add the eggs and egg yolk one at a time, beating on low speed after each addition until the yolk just disappears. Do not overmix, as this incorporates too much air. Stir in the sour cream, heavy cream, and vanilla extract until just combined.

For the swirl: In a small bowl, beat together the 4 oz softened cream cheese, powdered sugar, 1 tbsp heavy cream, and 1/2 tsp vanilla until smooth.

Pour the cheesecake filling slowly and evenly over the cooled carrot cake layer.

Drop spoonfuls of the cream cheese frosting mixture over the top of the cheesecake filling. Use a knife or skewer to gently swirl the frosting into the top layer of the batter for a marbled effect.

Place the springform pan into a large roasting pan. Carefully pour hot water into the roasting pan until it comes about 1 to 1.5 inches up the side of the springform pan.

Bake at 325°F (163°C) for 70-80 minutes. The edges of the cheesecake should be set, but the center 2-3 inches should still have a slight wobble (like Jell-O).

Turn off the oven and let the cheesecake cool in the oven with the door cracked for 1 hour. This prevents cracking.

Remove the cheesecake from the water bath and let it cool completely on a wire rack. Once at room temperature, cover loosely with plastic wrap and refrigerate for at least 6 hours, but preferably overnight.

Before serving, run a thin knife around the edge of the pan before releasing the springform side. Garnish with toasted chopped pecans. Slice with a hot, clean knife for best results.