01 - Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round or square baking pan to ensure easy release after baking.
02 - In a large bowl, whisk together flour, granulated sugar, brown sugar, baking powder, baking soda, salt, and cinnamon. In a separate bowl, whisk the eggs, vegetable oil, and buttermilk until well blended. Incorporate the wet mixture into the dry ingredients, stirring just until combined.
03 - Gently fold the finely grated carrots and drained crushed pineapple into the batter, ensuring they are evenly distributed without overmixing.
04 - In a medium bowl, beat the cream cheese until smooth. Add granulated sugar, vanilla extract, and egg, mixing until a uniform, creamy consistency is achieved.
05 - Pour the carrot batter into the prepared pan, smoothing the top. Gently spoon the cheesecake filling over the batter. In a small bowl, combine melted butter, brown sugar, and cinnamon; crumble this mixture evenly over the top.
06 - Place the pan into the oven and bake for 35 to 40 minutes, or until the center is set and a toothpick inserted near the middle emerges clean. Allow to cool completely before slicing and serving.