Preheat the oven to 350°F (175°C). Grease and line a 9×13-inch baking pan.
Make the carrot cake batter: In a bowl, whisk together flour, baking powder, baking soda, cinnamon, nutmeg, and salt. In another bowl, beat eggs with both sugars, oil, and vanilla until smooth. Stir in the grated carrots, then fold in the dry ingredients until just combined.
Prepare the cheesecake filling: Beat softened cream cheese with sugar, egg, and vanilla until silky and smooth.
Layer the cake: Spread half of the carrot cake batter in the prepared pan. Carefully pour the cheesecake mixture over the top, spreading it evenly. Spoon the remaining carrot cake batter over the cheesecake layer and smooth it out.
Add the honeybun swirl: Mix brown sugar and cinnamon together. Sprinkle generously over the top and use a knife to gently swirl it into the batter.
Bake: Place the pan in the center of the oven and bake for 50–55 minutes, or until the cake is set and a toothpick inserted comes out clean (avoid the cheesecake layer when testing).
Cool completely: Allow the cake to cool in the pan for 15 minutes, then transfer to a wire rack to cool for at least 1 hour before glazing.
Glaze: Whisk together powdered sugar, milk, and vanilla until smooth. Drizzle the glaze generously over the cooled cake.
Slice and serve: Cut into squares and enjoy the layers of moist carrot cake, creamy cheesecake, and sweet honeybun swirl.