Pour the apple cider into a saucepan and simmer it over medium-low heat. Let it cook down until only about half a cup remains, which usually takes 10 to 15 minutes. Set it aside to cool completely before adding it to the batter.
In a separate bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and cloves. Whisking evenly distributes the leavening and spices. This step only takes a minute but keeps your loaf from having pockets of baking soda.
Using a hand mixer, beat the softened butter with the granulated and brown sugars until light and fluffy. Add the eggs one at a time, mixing well after each addition. This creates the airy base your bread needs to rise properly.
Alternate adding the flour mixture and cooled cider into the butter mixture, starting and ending with the flour. Mix just until combined, since overmixing leads to a dense, tough loaf instead of a tender one.
Toss the chopped apples with lemon juice in a small bowl, then gently fold them into the batter using a rubber spatula. The lemon juice keeps the apples looking fresh and prevents excess browning inside the loaf.
Pour the batter into a greased and floured 9×5-inch loaf pan. Bake at 350°F for 55 to 60 minutes, checking with a toothpick for doneness. Let it cool in the pan for 15 minutes before transferring to a wire rack.
Whisk together confectioners' sugar, cinnamon, and milk until smooth and pourable. Once the loaf is completely cool, drizzle the glaze over the top with a spoon. Slice and serve once the glaze has set.