Place the butter and apples over medium heat in a heavy-bottomed skillet. Cook until the apples are tender, about 5 minutes. Deglaze with the apple cider and immediately add the brown sugar and cinnamon. Cook until the apples are soft, an additional 3 to 4 minutes. Set aside to cool to room temperature.
Add the ice cream, milk, half of the crushed peppermint candy and 4 shortbread cookies to a blender, along with the cooled apples. Blend until smooth. Pour the contents into 6 tall milkshake glasses and top each with more peppermint candy and a shortbread cookie.