Preheat the oven to 350°F (175°C) and grease a 9×5-inch loaf pan.
In a bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
Beat the softened butter, granulated sugar, and brown sugar until light and fluffy.
Add the eggs one at a time, mixing well after each addition.
Mix in the vanilla extract and applesauce.
Add the dry ingredients alternately with the apple cider, beginning and ending with the dry ingredients.
Mix just until the batter is combined; do not overmix.
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.
Let the loaf cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely.
For the glaze, whisk powdered sugar, apple cider, cinnamon, and vanilla until smooth.
Drizzle the glaze generously over the cooled loaf.
Allow the glaze to set for a few minutes before slicing and serving.