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

2. In a medium bowl, beat the softened cream cheese and sugar until smooth.

3. Add the egg and vanilla to the cream cheese mixture and mix just until creamy. Set aside.

4. In another bowl, whisk together flour, cinnamon, nutmeg, baking powder, baking soda, and salt.

5. In a large bowl, whisk vegetable oil, granulated sugar, brown sugar, eggs, and vanilla until smooth.

6. Stir the grated carrots into the wet mixture.

7. Fold the dry ingredients into the carrot mixture until just combined, then fold in nuts if using.

8. Spread about two-thirds of the carrot cake batter into the prepared pan.

9. Spoon the cheesecake mixture over the batter, then add small dollops of the remaining carrot cake batter on top.

10. Gently swirl the layers with a knife or skewer without overmixing.

11. Bake for 32 to 38 minutes, or until the edges are set and the center has only a slight jiggle.

12. Cool completely at room temperature, then refrigerate for at least 3 hours before slicing.

13. Lift the bars from the pan using the parchment paper, slice into squares, and serve chilled.