Preheat the oven to 350 degrees F (175 degrees C). Spray a 8-inch round cake pan with cooking spray.

Whisk together flour, white sugar, brown sugar, baking soda, baking powder, cinnamon, pumpkin pie spice, and salt in a large bowl.

Combine egg whites and vanilla extract in a small bowl. Beat using an electric mixer until fluffy.

Place carrots into a food processor and process until finely grated but not pureed into a pulp, 2 to 5 seconds.

Pour carrots and applesauce into flour mixture; mix well. Fold in egg white mixture and mix until thoroughly combined. Spoon batter into the prepared cake pan.

Bake in the preheated oven until a knife inserted into the center comes out clean and the cake is pulling away from the sides of the pan, 27 to 30 minutes. Remove from oven and let cake cool completely on a wire rack.