Preheat oven to 400 degrees. Line 2 baking sheets with parchment paper. In large bowl, beat cake mix, eggs, water and oil with an electric hand mixer, until well combined.
Scoop equal portions of batter onto prepared baking sheets. Spread batter with the back of a spoon into a circle.
Bake for 8 to 10 minutes, or until batter is fully set. Remove from oven and cool.
Meanwhile prepare Peppermint Cream Cheese Filling: In a large bowl, combine softened cream cheese, butter and peppermint extract. Beat with an electric mixer on medium speed until light and fluffy. Gradually beat in confectioners' sugar. If necessary, add milk (1 teaspoon at a time) to make a filling of spreading consistency.
Remove cooled cakes from parchment, matching them with another one that is similar in size. Spoon or pipe peppermint filling onto the flat side of one cake, top with another cake and sandwich together.
Serve cookies immediately. If chilled, let stand at room temperature for 15 minutes before serving. If desired, sprinkle each cookie with some finely chopped peppermint candies before serving or roll the edges in something fun like crushed candy canes or mini chocolate chips.
To Store: Layer cookies between sheets of waxed paper in an airtight container; cover. Store in the refrigerator up to 3 days.