Preheat the oven to 350°F. Line an 8×8 square baking pan with parchment paper and spray lightly with cooking spray.
Whisk flour, baking powder, baking soda, cinnamon, allspice, and salt together in large bowl. Set aside.
In bowl of standing mixer with paddle attachment (or using a hand mixer), beat both sugars, oil, and eggs on medium until for about 30 seconds.
Add carrots and remaining dry ingredients. Mix by hand will there is not visible flour.
Pour the batter into the cake pan and bake for 30-35 minutes.
Remove from oven and cool for at least an hour before frosting.