Preheat oven to 350 degrees F (175 degrees C). Grease 12 muffin cups or line with paper liners.
Mix sugar, applesauce, and butter together in a bowl.
Beat egg whites in a glass or metal bowl until medium peaks form. Lift your beater or whisk straight up: the tip of the peak formed by the egg whites should curl over slightly.
Stir egg yolks and vanilla extract into applesauce mixture; fold in egg whites. Stir in flour, baking powder, and salt until just combined. Beat coconut milk into batter until smooth, about 1 minute. Beat in shredded coconut for about 30 seconds. Pour batter into the prepared muffin cups.
Bake in the preheated oven until a toothpick inserted in the center of a cupcake comes out clean, about 20 minutes.