In a large mixing bowl, whisk the softened cream cheese until smooth. Add the granulated sugar and flour, mixing until well combined.

Incorporate the eggs one at a time, ensuring each is fully integrated before adding the next.

Whisk in the sour cream, vanilla extract, and a pinch of sea salt until the mixture is creamy and smooth. Set aside.

Preheat the oven to 350°F. Use a 9-inch springform pan. Grease the bottom and sides of the pan with butter or non-stick cooking spray. Cut a circle of parchment paper to fit the base of the pan and place it inside. Optionally, cut strips of parchment paper to line the sides of the pan. This preparation helps prevent the cheesecake from sticking and allows for a clean release.

In a medium bowl, whisk together the all-purpose flour, baking powder, sea salt, ground cinnamon, nutmeg, and ginger.

In a separate large bowl, with an electric mixer, mix the eggs, light brown sugar, and granulated sugar until well combined. Stir in the unsweetened applesauce, canola (or vegetable) oil, and vanilla extract.

Gradually add the dry ingredients to the wet mixture, mixing just until combined. Fold in the finely grated carrots.

Pour half of the carrot cake batter into the prepared springform pan, spreading it evenly.

Gently pour the cheesecake batter over the carrot cake layer. Spoon the remaining carrot cake batter over the cheesecake layer in small dollops.

Place the pan on the center rack of the preheated oven and bake for 70-80 minutes, or until the center is set. If the top begins to brown too quickly, tent it with aluminum foil.

Remove the cake from the oven and let it cool in the pan for 15 minutes. Run a knife around the edge to loosen it, then remove the sides of the springform pan. Allow the cake to cool to room temperature before refrigerating for at least 4 hours, or overnight, to set.

In a medium bowl, beat the butter and powdered sugar until light and fluffy. Gradually add softened cream cheese, mixing until smooth. Stir in the vanilla extract.

In a skillet over medium heat, melt the unsalted butter. Add the granulated sugar and chopped pecans, stirring continuously until the sugar dissolves and coats the pecans, and they become fragrant and toasted, about 3–5 minutes. Transfer the pecans to a sheet of parchment paper to cool.