Preheat the oven to 350°F. Line (2) 12-cup muffin tins with cupcake liners; set aside.
Finely chop ¾ cup of the pecans. Reserve the remaining ¼ cup of whole pecans for garnish; set aside.
In a large mixing bowl, whisk together the eggs and vegetable oil. Add the brown sugar, white sugar, molasses, applesauce, and vanilla extract, and continue whisking until combined.
In a second mixing bowl, combine the flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves. Stir to combine.
Gradually add small amounts of the dry ingredients to the wet and whisk to combine. Repeat until all the dry ingredients have been mixed into the batter, then stir in the chopped pecans.
Add the grated carrots and fold into the mixture thoroughly.
Fill each cupcake liner with the batter until approximately ¾ full - do not overfill. Place the muffin pans into the pre-heated oven for 18-20 minutes, or until a toothpick inserted in the center comes out clean. Remove from oven and transfer the cupcakes to wire racks to cool completely.
While the cupcakes are cooling, prepare the frosting by beating together the cream cheese, butter, vanilla extract, and salt. Slowly add the powdered sugar a little at a time until thoroughly combined into the frosting. Set aside.
Once cupcakes are completely cool, frost with the cream cheese frosting and garnish with the remaining whole pecans, if desired.