Apple Cider Cookies

Name: Apple Cider Cookies
Cuisine: American
Ingredients: 3 cups apple cider; 2 sticks (1 cup) unsalted butter; 3 cups all-purpose flour; 1/2 teaspoon baking powder; 1/2 teaspoon baking soda; 1½ teaspoons cinnamon; 1/4 teaspoon ground nutmeg; 1/8 teaspoon ground cloves; 1¼ teaspoons salt; 1 cup packed light brown sugar; 1/2 cup granulated sugar; 1 large egg, room temperature; 1 large egg yolk, room temperature; 1½ teaspoon vanilla extract; 1/3 cup apple cider reduction; 1 small apple, peeled and cubed (about 1/2 cup); 2 tablespoons unsalted butter; 1 tablespoon apple cider reduction; 2 teaspoons cinnamon; 1 cup powdered sugar; 1/4 teaspoon vanilla extract; 1½ tablespoons whole milk