In a mixing bowl, beat the cream cheese until smooth. Add the sugar and flour, and mix well. Then, add the eggs one at a time, ensuring each is mixed in thoroughly. Finally, add the sour cream, vanilla extract, and a pinch of sea salt. Mix until smooth and creamy.

In another bowl, combine the flour, baking powder, sea salt, cinnamon, nutmeg, and ginger. In a separate bowl, whisk together the eggs, brown sugar, granulated sugar, applesauce, oil, and vanilla extract. Gradually add the flour mixture to the wet ingredients and mix until combined. Fold in the finely grated carrots.

Preheat your oven to 350°F (175°C). Grease a springform pan. Pour half of the carrot cake batter into the pan, followed by the cheesecake batter, and finally the remaining carrot cake batter on top.

Bake in the preheated oven for about 60-70 minutes, or until the center is set and not jiggly. Let it cool in the pan for 10 minutes before running a knife around the edges and removing the sides.

Allow the cheesecake to cool completely at 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 the frosting is fluffy.

For added flavor, you can caramelize some pecans in a skillet with a bit of sugar and butter until golden and crispy.

Once chilled, frost the cheesecake with cream cheese frosting and garnish with the caramelized pecans if using. Slice and serve chilled.