Pour the apple cider into a small saucepan and bring it to a gentle simmer over medium heat.
Cook for 10–15 minutes, until the cider has reduced from 1 cup to approximately ½ cup.
Remove from the heat and allow the reduced cider to cool completely.
Preheat the oven to 375°F (190°C). Line a 12-cup muffin pan with paper liners or lightly grease it.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a large bowl, beat softened butter and sugar for 2–3 minutes, until light and fluffy.
Add the eggs one at a time, mixing well after each addition.
Stir in the vanilla extract.
Gradually mix in the cooled reduced apple cider.
Add the dry ingredients and gently stir until just combined. Do not overmix.
Fold in the finely diced apple.
Divide the batter evenly between the muffin cups, filling each about ¾ full.
Mix the sugar and cinnamon for the topping in a small bowl.
Lightly brush the muffin tops with melted butter and sprinkle generously with the cinnamon-sugar mixture.
Bake for 18–22 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the muffins cool in the pan for 5 minutes, then transfer them to a wire rack.
Serve warm or allow them to cool completely before storing.
