Preheat the oven to 350°F (175°C) and grease a 9×5-inch loaf pan or line it with parchment paper.
Whisk together gluten-free 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.
Stir in the vanilla extract and applesauce.
Add the dry ingredients alternately with the apple cider, mixing just until the batter is smooth.
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 10–15 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 over the completely cooled loaf.
Let the glaze set for a few minutes before slicing and serving.