Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper.
In a large bowl, cream together the softened butter and packed brown sugar until light and fluffy.
Beat in the egg and vanilla extract.
Stir in the grated apple until well combined.
In a separate bowl, whisk together flour, baking soda, baking powder, cinnamon, nutmeg, cloves, and salt.
Gradually mix the dry ingredients into the wet ingredients until just combined.
Fold in the chopped nuts, if you are using them.
Drop spoonfuls of dough (about 1 1/2 tablespoons) onto the prepared baking sheets, spacing them 2 inches apart.
Bake for 10–12 minutes, or until the edges are golden and the centers are set.
Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.
Beat together the cream cheese, softened butter, powdered sugar, vanilla, and milk until smooth and creamy.
Spread or drizzle the cream cheese icing over the cooled cookies.
Let the icing set before serving.