1. Prepare the Oven and Pan:

1. Preheat oven to 350°F (175°C).
2. Spray an 8×8-inch baking pan with non-stick spray. For easy removal, line the pan with parchment paper, leaving an overhang on two sides, and then lightly spray the parchment paper with non-stick baking spray.

2. Make the Carrot Cake Batter:

1. Melt the butter in a microwave-safe bowl. Microwave for 30 seconds, and then in 15-second intervals if needed, until completely melted.
2. In a medium bowl, combine the melted butter and brown sugar. Mix well with a spatula or whisk until combined.
3. Stir in the egg and vanilla extract until well combined.
4. In a separate bowl, whisk the flour, cinnamon, baking powder, and salt.
5. Add the dry ingredients to the wet ingredients and stir until just combined. Do not overmix.
6. Fold in the shredded carrots until evenly distributed.

3. Make the Cheesecake Swirl:

1. In a separate medium bowl, using a hand mixer (or stand mixer), beat the softened cream cheese and granulated sugar together until smooth and creamy (about 1-2 minutes).
2. Add the egg yolk and vanilla extract. Beat until well combined and smooth.

4. Assemble the Bars:

1. Spread about half of the carrot cake batter evenly into the prepared baking pan.
2. Drop dollops of half of the cheesecake batter over the carrot cake layer. Do not spread the cheesecake batter at this point.
3. Spread the remaining carrot cake batter over and around the cheesecake dollops.
4. Drop dollops of the remaining cheesecake batter over the top of the carrot cake layer.
5. Using a knife or skewer, swirl the cheesecake batter into the carrot cake batter, creating a marbled effect. Don’t over-swirl, or the colors will become muddy.

5. Bake:

1. Bake for 35-40 minutes, or until the edges are golden brown and a toothpick inserted into the center (avoiding a cheesecake swirl) comes out with a few moist crumbs attached. There should be very little jiggle in the center.

6. Cool and Chill:

1. Remove the pan from the oven and let the bars cool completely in the pan.
2. Once cooled, cover the pan with plastic wrap and refrigerate for at least 2 hours, or preferably longer, to allow the cheesecake swirl to fully set.

7. Serve:

1. Lift the bars out of the pan using the parchment paper overhang (if used) or carefully cut them into squares within the pan.
2. Serve chilled.