Preparing the Carrot Cake:

1. Preheat Oven: Set to 350°F. Line the bottom of an 8-inch or 9-inch springform pan with parchment paper and grease it.
2. Mix Dry Ingredients: In a bowl, whisk together flour, baking powder, baking soda, salt, and cinnamon.
3. Combine Wet Ingredients: In a large bowl, whisk together vegetable oil, granulated sugar, brown sugar, eggs, and vanilla extract until smooth.
4. Mix Batter: Add dry ingredients to the wet mixture and mix on low speed until just combined. Fold in the grated carrots.

Preparing the Cheesecake Batter:

1. Beat Cream Cheese: In a large mixing bowl, beat cream cheese for 1 minute until smooth.
2. Add Sugars: Mix in brown and granulated sugar, beating for another minute. Scrape the bowl and beat for an additional 30 seconds.
3. Add Sour Cream: Mix in sour cream until combined.
4. Mix Eggs and Vanilla: Add eggs one at a time, mixing until just incorporated. Add vanilla extract and salt with the last egg.

Assembling the Cake:

1. Layer Batters: Pour 1/3 of the carrot cake batter into the prepared pan. Follow with 1/3 of the cheesecake batter in the center. Repeat with another 1/3 of each batter, then finish with the remaining carrot cake batter and cheesecake batter.
2. Bake: Bake for about 50 minutes, until the center is slightly jiggly and edges are set. Cover with foil if edges brown too quickly.
3. Cool in Oven: Turn off the oven, prop the door open, and leave the cheesecake inside for 20 minutes to prevent sinking.
4. Chill: Cool completely, then refrigerate for at least 6 hours or overnight. Remove the pan ring once fully chilled.

Making the Caramelized Pecans:

1. Cook Pecans: In a small saucepan over medium heat, melt butter and sugar. Add chopped pecans and stir for about 4 minutes until caramelized.
2. Cool: Spread on parchment paper to cool.

Making the Cream Cheese Frosting:

1. Beat Butter and Cream Cheese: In a medium bowl, beat cream cheese and butter for 30 seconds.
2. Add Sugar and Vanilla: Mix in powdered sugar and vanilla extract until smooth.

Decorating the Cake:

1. Frost Cake: Spread Cream Cheese Frosting over the chilled cheesecake using a spatula.
2. Add Pecans: Top with caramelized pecans.