Preheat the oven to 425°F (218°C). Grease a 12-cup muffin pan or line it with paper liners.
Prepare the crumb topping by mixing brown sugar, granulated sugar, and cinnamon. Stir in melted butter, then gently mix in the flour with a fork until large crumbs form. Set aside.
In a large bowl, whisk together flour, baking soda, baking powder, cinnamon, and salt.
In another bowl, beat the softened butter with both sugars until light and creamy, about 3 minutes.
Add eggs, yogurt, and vanilla extract. Beat until mostly smooth.
Gradually mix the dry ingredients into the wet ingredients, alternating with the milk. Mix until just combined.
Fold in the chopped apples without overmixing.
Divide the batter evenly among the muffin cups, filling each nearly to the top.
Sprinkle a generous amount of crumb topping over each muffin, gently pressing it into the batter.
Bake the Apple Cinnamon Crumb Muffins for 5 minutes at 425°F (218°C). Without opening the oven, reduce the temperature to 350°F (177°C) and continue baking for 15–18 minutes, or until a toothpick inserted in the center comes out clean.
Cool in the pan for 5 minutes before transferring to a wire rack.
For the icing, whisk together confectioners' sugar, cream, and vanilla until smooth. Drizzle over warm or cooled muffins.