Prepare the Oven and Pan: Preheat the oven to 425°F (218°C). Spray a 12-count muffin pan with nonstick spray or line it with muffin liners. Since the recipe makes up to 14 muffins, prepare a second pan if needed, or plan to bake in batches using one pan.
Make the Crumb Topping: In a medium bowl, combine the brown sugar, granulated sugar, and cinnamon. Stir in the melted butter, then gently work in the flour with a fork. Keep the mixture in large, loose crumbles and avoid over-mixing; excessive mixing can turn the topping into a thick paste. Set aside. Combine the Dry Ingredients: Whisk the flour, baking soda, baking powder, cinnamon, and salt together in a large bowl until evenly combined. Set the dry mixture aside.
Cream the Butter and Sugars: Using a handheld or stand mixer fitted with a paddle attachment, beat the softened butter, brown sugar, and granulated sugar on high speed for about 3 minutes, until smooth and creamy. Scrape the sides and bottom of the bowl as needed. Add the Wet Ingredients: Add the room-temperature eggs, yogurt or sour cream, and vanilla extract to the creamed mixture. Beat on medium speed for 1 minute, then increase to high speed and continue until the mixture is combined and mostly creamy. A slightly curdled appearance is fine. Scrape the bowl again as needed. Finish the Batter: With the mixer on low speed, gradually add the dry ingredients and milk to the wet mixture. Mix just until no pockets of flour remain. Fold in the chopped apples, distributing them evenly throughout the batter.
Fill the Muffin Pan: Divide the batter evenly among the prepared muffin cups, filling each one all the way to the top. Spoon the crumb topping over each muffin and gently press the crumbs down so they adhere to the batter.
Begin Baking: Place the muffins in the 425°F (218°C) oven and bake for 5 minutes.
Reduce the Temperature: Without removing the muffins from the oven, reduce the oven temperature to 350°F (177°C). Continue baking for another 15–18 minutes.
Check for Doneness: Bake until a toothpick inserted into the center of a muffin comes out clean. The total oven time is about 20–23 minutes.
Cool the Muffins: Let the muffins cool in the pan for 5 minutes, then transfer them to a wire rack to continue cooling.
Prepare the Vanilla Icing: Whisk the confectioners’ sugar, heavy cream or milk, and vanilla extract together until smooth.
Drizzle and Serve: Drizzle the icing over the warm or cooled muffins. The muffins can also be served plain without icing.
Store the Muffins: Keep iced or plain muffins covered at room temperature for a few days, or refrigerate them for up to 1 week.
Freeze for Longer Storage: For best results, freeze the baked and cooled muffins without icing for up to 3 months. Thaw them in the refrigerator or at room temperature, then add fresh icing before serving.