Cook the Apples: Melt the butter in a 12-inch skillet over medium-high heat. Add the apples, brown sugar, and cinnamon. Cook, stirring often, until the moisture has completely evaporated and the apples are well browned, 8 to 10 minutes. Remove the pan from the heat and allow to cool for 10 minutes.
Prepare the Muffins: Adjust an oven rack to the upper-middle position and preheat the oven to 400 degrees F. Line a 12-cup muffin tin with paper liners or spray with non-stick cooking spray.
In a large bowl, whisk together the flour, baking powder, baking soda, salt and ground cinnamon.
In a medium bowl, whisk together the granulated sugar, eggs, oil, and melted butter until thick, about 30 seconds. Whisk the apple cider, yogurt and vanilla into the sugar mixture until combined.
Fold the sugar mixture and the cooled apples into the flour mixture until just combined. Divide the batter evenly among the prepared muffin cups (there should be about ⅓ cup of batter per cup, and the cups will be filled to the rim).
Mix Together the Topping: In a small bowl, stir together the sugar, brown sugar and cinnamon for the topping. Sprinkle the muffin tops evenly with the topping.
Bake: Bake until golden brown and a toothpick inserted in the center comes out with a few moist crumbs attached, 18 to 22 minutes, rotating the muffin tin halfway through baking. Let the muffins cool in the muffin tin on a wire rack for 10 minutes. Remove the muffins from the pan and place on a wire cooling rack. Cool for 5 more minutes before serving. The muffins can be served warm or at room temperature. Store the muffins in an airtight container at room temperature for up to 3 days.
Apple Cider: Substitute equal amount of apple juice or other fruit juice (white grape juice is a great neutral-flavored option).
Yogurt: Substitute an equal amount of sour cream or whole buttermilk. If using low-fat buttermilk, then only use ¼ cup + 2 tablespoons
Add-ins & toppings: See suggested substitutions, combinations, and instructions in the post above.
Mini muffins: This recipe will yield 24 mini muffins; bake for 10 to 14 minutes.
Quick bread: You can use this batter to bake a quick bread. Spread it in a 9x5-inch loaf pan and bake for 55 to 60 minutes.
Storage: Store the muffins in an airtight container at room temperature for up to 3 days or in the refrigerator for up to 1 week.
Freezing: Once completely cool, the muffins can be wrapped in plastic wrap and transferred to an airtight container or zip-top bag and frozen for up to 3 months. Thaw at room temperature.
Reheat - Reheat in the oven (or toaster oven) at 350 degrees F, wrapped in foil, for 15 to 25 minutes. Reheat in a microwave - place on a plate, cover with a damp paper towel, and heat for 15 to 30 seconds.