Preheat Oven: Preheat your oven to 350°F (175°C). Grease a 9×13-inch baking dish with cooking spray or line it with parchment paper, leaving some overhang for easy lifting.
Make the Dry Mix: In a large bowl, whisk together the flour, granulated sugar, brown sugar, baking soda, baking powder, salt, cinnamon, nutmeg, and cloves. Set aside.
Prepare the Wet Ingredients: In a separate bowl, whisk together the oil, sour cream, eggs, and vanilla extract.
Combine: Add the wet ingredients to the dry ingredients and mix until just combined. Fold in the shredded carrots, being careful not to over-mix.
Prepare the Cheesecake Batter: In another large bowl, use an electric hand mixer to beat together the cream cheese, sugar, and vanilla until smooth. Add the eggs one at a time, beating after each addition until fully blended.
Layer the Batters: Spread half of the carrot cake batter into the prepared baking dish. Spoon dollops of the cheesecake batter on top, followed by the remaining carrot cake batter. Use a knife to gently swirl the batters together to create a marbled effect. Don’t over-swirl.
Bake: Bake the bars in the preheated oven for 35-40 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Cool and Serve: Allow the bars to cool on a wire rack. Once cooled, cover and refrigerate. These bars are best served chilled. Store leftovers covered in the refrigerator for up to 1 week.