Preheat the oven to 375 degrees F. Grease or line the muffin tins with paper liners
Make the streusel. Combine the brown sugar, flour, and cinnamon. Add the 1 tablespoon butter and cut in until the ingredients are well combined. Toss in the walnuts. Set this aside.
In a large bowl combine the flour, sugar, baking powder, baking soda, and salt.
In a different bowl add the eggs, 1/2 cup butter, milk, and vanilla.
Add the wet ingredients to the dry and mix just until combined. Fold in the apples
Divide the batter between 12 prepared muffin cups. The muffin cups should be around 2/3 to 3/4 full.
Place in the preheated oven and bake for 18 to 20 minutes or until a wooded pick inserted in the center comes out clean or with a few moist crumbs.
Cool the muffins in the tin for about 5 minutes, then remove them from the tins and set them on a cool rack. Cool completely.
Once the muffins are cooled, make the icing. Combine the powdered sugar, 1 teaspoon butter, salt, and vanilla. Beat in the milk until you have the right consistency to drizzle over top of the muffins.