Preheat oven to 350*F. Line muffin tin with paper liners and spray with non-stick cooking spray. (if using more than one cupcake tray, be sure to fill any empty muffin tins with water, about ¼ full)
In a medium bowl mix all the dry ingredients and set aside.
In a large bowl whisk the Greek yogurt, applesauce, and maple syrup until well combined. Add the egg and vanilla and continue whisking until combined.
Add the dry ingredients and grated carrots to the wet ingredient bowl. Whisk until just combined, being careful not to over mix. With an ice cream scoop or spoon, divide the batter among the muffin cups, filling each muffin cup 3/4 full. Bake for 20-23 minutes until toothpick inserted comes out clean and top 'gives back' when touched.
Remove pan from oven and let muffins cool for two minutes. Transfer to a wire rack to cool completely. Be sure to cool completely before frosting.
While cupcakes are cooling prepare the frosting. Place all ingredients into a bowl. Using an electric mixer or stand mixer, beat until well combined and fluffy. (Taste, can always add more powdered sugar if you like it more sweet. Or if it is too thick you can thin with a splash of orange juice.)
Frost cupcakes! You will have a little extra frosting leftover depending upon how much icing you put on each. Store in an airtight container in the refrigerator.