Preheat your oven to 350°F (175°C). Grease or line the 8×8 pan. Melt the butter in short bursts, then whisk in brown sugar. Add the egg and vanilla, stirring until smooth. Then mix in flour, cinnamon, baking powder, and salt; fold in the shredded carrots so the batter is just combined.
In the second bowl, beat the cream cheese with sugar until creamy. Add the egg yolk and vanilla, then beat until smooth and fully incorporated. Pause if needed to scrape the sides.
Pour about half of the carrot cake batter into the prepared pan and smooth it down. Drop dollops of the cheesecake mix over it, then add the remaining carrot batter on top. Finally, place more dollops of cheesecake and swirl them gently with a skewer or butter knife—don’t overmix.
Bake for 35–40 minutes, until edges are golden and the center has just a tiny bit of jiggle. Let the bars cool completely in the pan before slicing them. Refrigerate afterward to help them set fully.