With the rack in the middle position, preheat the oven to 350°F (180°C). Butter an 8-inch (20 cm) square pan and line with a strip of parchment paper, letting it hang over two sides.
In a bowl, combine the flour, baking powder, baking soda and cinnamon.
In another bowl, whisk the oil, sugar and brown sugar with an electric mixer. Add the eggs and whisk until smooth. Using a wooden spoon, add the carrots and pineapple. Add the dry ingredients. Pour into the prepared pan.
Bake for 45 minutes or until a toothpick inserted in the centre of the cake comes out clean. Let cool on a wire rack. Unmould and let cool completely, for approximately 2 hours.
Cream Cheese Icing
In a bowl, whisk the cream cheese, butter and vanilla with an electric mixer. Gradually add the icing sugar and whisk until smooth and creamy. Spread the icing over the cake. Garnish with carrot ribbons, if desired.
The cake will keep for 5 days in the refrigerator. Let sit out at room temperature before serving.