Spray an 8x8 baking dish with cooking spray and set aside. In large mixing bowl, stir together melted butter and brown sugar. Add egg, vanilla, flour, cinnamon, baking powder, and salt and stir again, by hand, until well combined and smooth. Add in shredded carrots and stir until incorporated. In a mixing bowl, place cream cheese, sugar, egg, and vanilla. Beat with an electric mixer until smooth and creamy. Place half of the carrot cake batter into the pan. Drop spoonfuls of 1/2 of the cheesecake batter over the top. Don't swirl just yet! Drop spoonfuls of the rest of the carrot cake batter on top of this and then follow with the remaining cheesecake batter. Take a knife and swirl all of this together a bit over the top. Bake at 350 for 35-40 minutes or until it is lightly browned and set in the center. Remove and allow to cool before serving for the prettiest pieces.
Now y'all know I don't generally fuss over recipes so having you put one egg yolk in the cream cheese mixture may seem petty, but it makes a difference. If you use the whole egg your mixture will be runny. It will still work, but you won't be able to swirl it as well.