Preheat the oven to 350° F. Line a 9 inch square baking pan with parchment paper. To make the carrot cake. In a mixing bowl, beat together the melted butter, brown sugar, eggs, and vanilla until combined. Add the flour, baking powder, baking soda, salt, and cinnamon, beating until just combined. Stir in the grated carrots.To make the cheesecake. In a mixing bowl, beat together the cream cheese, sugar, and flour until whipped, about 3 minutes. Add the eggs, one at time until incorporated. Add the vanilla and beat once more until the mixture is smooth. Assemble. Working with the carrot cake batter, spread about 2/3rds of the batter into the prepared pan in an even layer. Now pour over 2/3rds of the cheesecake batter, then dollop spoonfuls of the remaining carrot cake batter into the cheesecake layer. Lastly, pour/drizzle over the remaining cheesecake batter. Take the pan and tap it against the counter 2-3 times to help pop any air pockets in the cheesecake. Bake for 45-55 minutes, tenting the cake with foil halfway through cooking to prevent the top from browning. Allow the cake to cool in the pan, then transfer to the fridge to chill completely, at least 3 hours.To make the icing, whisk the butter, powdered sugar, vanilla, and 2 tablespoons water (or milk), until smooth. If needed, thin with additional water/milk. Drizzle the icing over the bars. Let set for 1-ish hour...or slice, snack, and enjoy! The bars can keep chilled in the fridge for up to 5 days.