Melt the butter in a medium skillet over medium heat.

Add the diced apples, brown sugar, granulated sugar, lemon juice, cinnamon, nutmeg, and salt.

Cook for about 5–7 minutes, stirring occasionally, until the apples begin to soften and release their juices.

Sprinkle in the cornstarch and stir until the apple mixture thickens slightly, about 1–2 minutes.

Remove from the heat and stir in the vanilla extract.

Set aside to cool slightly while preparing the muffin batter.

In a medium bowl, combine the flour, granulated sugar, brown sugar, cinnamon, and salt.

Pour in the melted butter.

Mix with a fork until large crumbs form.

Set aside.

Preheat the oven to 350°F.

Line a 12-cup muffin pan with cupcake liners or lightly grease each cavity.

In a large bowl, whisk together the flour, baking powder, baking soda, salt, granulated sugar, brown sugar, cinnamon, and nutmeg.

In a separate bowl, whisk together the buttermilk, vegetable oil, eggs, and vanilla extract.

Pour the wet ingredients into the dry ingredients.

Gently fold together just until combined. Do not overmix.

Divide the muffin batter evenly between the prepared muffin cups, filling each about ⅔ full.

Spoon a generous amount of the cooked apple mixture directly on top of each portion of muffin batter.

Sprinkle the crumble topping generously over the apples, covering the tops of each muffin.

Bake at 350°F for 18–22 minutes, or until the muffins are golden and a toothpick inserted into the muffin portion comes out clean or with a few moist crumbs.

Allow the muffins to cool in the pan for about 10 minutes.

Transfer to a wire rack and allow them to cool slightly before glazing.

Whisk together the powdered sugar and milk until smooth.

If the glaze is too thick, add a small splash of milk. If it is too thin, add additional powdered sugar.

Drizzle the glaze over the cooled or slightly warm muffins.