Preheat oven to 425?F (218?C). Spray a 12-count muffin pan with nonstick spray or line with cupcake liners.

Make the crumb topping: Mix the brown sugar, granulated sugar, and cinnamon together in a medium bowl. Stir in the melted butter, and then gently mix in the flour. Set topping aside.

Make the muffins: Whisk the flour, baking soda, baking powder, cinnamon, and salt together in a large bowl. Set aside.

Using a handheld or stand mixer, beat the butter and both sugars together on high speed until smooth and creamy, about 3 minutes. Add the eggs, yogurt, and vanilla extract. Beat on medium speed for 1 minute, then increase to high until mostly creamy.

With the mixer running on low speed, add the dry ingredients and milk into the wet ingredients and beat until no flour pockets remain. Fold in the apples.

Spoon the batter into each muffin cup, filling each to the top. Spoon crumb topping on each, gently pressing it down.

Bake for 5 minutes at 425?F, then reduce temperature to 350?F (177?C) and bake for an additional 15?18 minutes or until a toothpick inserted in the center comes out clean.

Allow the muffins to cool for 5 minutes in the muffin pan, then transfer to a wire rack.

Make the icing: Whisk all icing ingredients together and drizzle over warm or cooled muffins.