Preheat the oven to 350°F. Line cupcake pan with 12 cupcake liners and set aside.
Sift all dry ingredients (all-purpose flour, baking soda, baking powder, salt, all spice, and cinnamon) together and set aside.
In a large mixing bowl, whisk together vegetable oil, brown and granulates sugar until well combined, add eggs, unsweetened applesauce, and vanilla extract and whisk it again until smooth consistency.
Add dry ingredients to the mixture and mix it till well combined.
Add shredded carrots and chopped pecans and fold it in.
Divide the batter between 12 cupcake liners and bake for 20 minutes at 350°F. When they are ready, take them out of the oven and set aside to cool off for a few minutes, then transfer to a wire rack.