In a medium mixing bowl, add the egg, coconut oil, applesauce and yogurt. Beat together using a hand-held mixer or whisk.
In a large mixing bowl, using a whisk, blend together the flour and baking soda.
Add the wet ingredients (applesauce, egg, yogurt, coconut oil) to the dry ingredients (flour, baking soda) and mix together using a hand-held mixer or large spoon. Mix until just batter forms - it will be thick.
Add the shredded carrots and fold in.
Spoon the pupcake batter into the muffin cups, about 3/4 the way full.
Place in the preheated oven and bake for 20 minutes until a toothpick inserted in the middle comes out clean. If not, bake another 3-5 minutes.
Remove pupcakes from the pan and cool at room temperature on a wire rack while you make the frosting. If you used cupcake liners, remove them while cupcakes are still slightly warm to prevent sticking.