Preheat oven to 400 degrees. If making regular sized muffins, line 18 muffin cups with liners. If making mini muffins, line 48 muffins cups with liners (or spray with nonstick cooking spray).
In a large skillet over medium heat, add the apples, butter, sugar, cinnamon, and water and cook until tender (this will take 7-10 minutes). Add 1/3 cup flour and mix until all the apples are coated.
In another bowl, mix together your dry ingredients (remaining flour, baking powder, baking soda, cinnamon and salt).
In a separate bowl, mix the applesauce, sugar and brown sugar until smooth and creamy. Add in eggs, mixing after each egg is added, and then stir in vanilla.
Carefully stir the dry ingredients into the wet mixture, alternating pouring in the milk, until they are both completely added. Slowly fold in the apples.
Fill each muffin cup about 2/3 full and bake (regular muffins - 15-17 minutes; mini muffins - 10-12 minutes). Let muffins cool in their tin for 5 minutes, then remove onto wire rack and let cool.
While muffins cool, mix together glaze ingredients in a bowl. Dip each muffin top into the glaze and let glaze run down the sides of the muffin.