Preheat oven to 350 degrees. Line muffin tin with papers.
Add butter and sugar to a large mixing bowl and beat until light and fluffy.
Beat in the egg and vanilla.
Stir in the applesauce.
Beat in the flour and baking soda until just combined.
Spoon mixture into the prepared muffin tin, filling 2/3 of the way full.
To prepare the streusel, add the sugar, flour, and cinnamon to a medium bowl and stir to combine.
Add the butter and use a fork or pastry cutter to cut the butter into the flour until the mixture is made of coarse crumbs.
Sprinkle the mixture evenly over all of the muffins.
Bake for 22 minutes or until a tester comes out clean.
When muffins are cool, whisk together the powdered sugar and milk and drizzle over the tops.