1. Preheat the oven to 350°F and line an 8×8-inch baking pan with parchment paper, leaving overhang on the sides for easy lifting.

2. In a medium bowl, whisk together the flour, baking powder, cinnamon, nutmeg, and salt.

3. In a large bowl, whisk together the brown sugar, 1/4 cup of the granulated sugar, vegetable oil, 2 eggs, and 1/2 teaspoon vanilla extract until smooth.

4. Stir in the grated carrots, then fold in the dry ingredients just until combined.

5. In a separate bowl, beat the softened cream cheese with the remaining 1/4 cup granulated sugar, 1 egg, and remaining 1/2 teaspoon vanilla extract until smooth and creamy.

6. Spread most of the carrot cake batter into the prepared pan, reserving a few spoonfuls for the top.

7. Pour the cheesecake mixture over the carrot cake layer and spread it gently into an even layer.

8. Drop the reserved carrot cake batter over the cheesecake layer in small spoonfuls, then lightly swirl with a knife.

9. Bake for 32 to 38 minutes, until the center is set and a toothpick inserted into the carrot cake portion comes out with a few moist crumbs.

10. Cool completely, then refrigerate for at least 2 hours before slicing into bars.