In a large mixing bowl, beat 1/2 cup unsalted butter with an electric mixer on medium to high speed for 30 seconds until creamy. Add the white sugar, 2/3 cup packed brown sugar, cinnamon, baking soda, nutmeg, and cloves. Beat until the mixture is well combined, light, and fluffy. This creates a fragrant, spiced base for the cookies.

Beat in the egg and apple cider or apple juice until the mixture is smooth and well combined. This step incorporates moisture and helps bind the cookie dough.

Add the flour and finely chopped peeled apple to the dough. Beat on low speed or stir until just combined, being careful not to overmix. The apple pieces add a burst of sweetness and moisture to every bite of cookie. I like to gently fold in the apples at the end to help keep their shape.

Drop the dough by rounded teaspoons, spacing them about 2 inches apart, onto a prepared cookie sheet lined with parchment paper or lightly greased. Bake in a preheated 375°F (190°C) oven for 10 to 12 minutes, until the edges are light brown and the tops spring back lightly when pressed. Let the cookies cool on the sheets for 1 minute, then transfer them to wire racks to cool completely.

In a medium saucepan, combine the softened butter, heavy cream, 1 cup brown sugar, and a pinch of salt. Cook over medium heat, stirring constantly, until the mixture comes just to a boil. Remove from the heat and stir in the vanilla extract. Let the mixture cool for a few minutes, then whisk in the powdered sugar gradually until the frosting reaches a smooth, spreadable consistency.

Once the cookies have cooled completely (from Step 4), spread the brown sugar frosting (from Step 5) generously over the tops. If you like, sprinkle a little freshly grated nutmeg or a pinch of sea salt on top for extra flavor. Store the finished cookies in an air-tight container in the refrigerator to keep them moist and delicious. I often like to add a bit of nutmeg on top—it gives the cookies an extra-special bakery touch!