Preheat oven to 425°F and prepare a 6-cavity muffin pan with nonstick liners. Set aside.
From the muffin ingredients, whisk together the flour, baking powder, baking soda, cinnamon, and salt in a large bowl. Set aside.
Using a stand mixer fitted with a paddle attachment, whip the butter and sugars together on high speed until creamy. Add the eggs, sour cream, and vanilla extract, and continue to mix until combined. Add the bowl of dry ingredients and milk to the wet ingredients and mix until a smooth batter forms. Stir in the apples and set aside.
In a medium mixing bowl, combine the crumble topping ingredients. Stir together the sugars, flour, and cinnamon. Stir in the melted butter, and then gently mix with your fingers just until a crumble forms. Set topping aside.
Fill the muffin liners equally and fully with muffin batter. Top with the crumble and gently press down on the tops to pack down the crumble.
Bake at 425°F for 5 minutes, reduce the oven temperature to 350°F, and continue baking for an additional 20-25 minutes or until a toothpick can be inserted into the middle of a muffin and come out clean. Let cool on the counter before removing from the muffin tin and adding the cinnamon sticks as an apple stem-style garnish.