Preheat the oven to 350°F (175°C) and grease a 9×5-inch loaf pan.
Whisk together flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
Beat the softened butter and sugar until light and fluffy.
Add the eggs one at a time, mixing well after each addition.
Stir in the vanilla extract and applesauce.
Add the dry ingredients alternately with the apple cider, mixing just until combined.
Pour the batter into the prepared loaf pan and smooth the top.
Bake for 45–50 minutes, or until a toothpick inserted into the center comes out clean.
Cool the loaf in the pan for 10 minutes, then transfer it to a wire rack.
Brush the warm loaf generously with melted butter.
Combine the sugar and cinnamon in a shallow dish.
Roll or sprinkle the cinnamon-sugar mixture over the buttered loaf until evenly coated.
Let the bread cool slightly before slicing and serving.