Simmer the apple cider in a saucepan over medium-low heat for 10–15 minutes, until it reduces to approximately ½ cup. Let it cool completely.

Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan or an 8-inch square cake pan and line the bottom with parchment paper.

Whisk together the flour, baking powder, cinnamon, nutmeg, and salt in a medium bowl.

Beat the softened butter and sugar in a large bowl for 2–3 minutes until pale and fluffy.

Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract.

Combine the cooled reduced apple cider with the milk.

Gradually add the dry ingredients to the butter mixture, alternating with the cider and milk mixture. Mix gently until just combined.

Pour the batter into the prepared pan and smooth the top evenly.

Bake for 30–40 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.

Cool the cake in the pan for 10–15 minutes, then carefully transfer it to a wire rack.

Mix the coating sugar and cinnamon in a shallow bowl.

Brush the slightly warm cake with melted butter, then sprinkle or gently roll it in the cinnamon-sugar mixture until evenly coated.

Slice and serve warm or at room temperature. Store leftovers in an airtight container.