Grease a 15 x 10-inch jellyroll pan; line with greased paper.
Sift flour, cocoa and baking soda into small bowl of electric mixer; add sugar, butter and water. Beat on medium speed for two minutes. Add egg; beat until combined. Spread mixture into prepared pan. Bake at 350 degrees F for about 20 minutes; cool in pan.
Peppermint Filling: Combine confectioners' sugar and oil in a small heatproof bowl; stir in enough milk to make a stiff paste. Add peppermint extract. Stir over hot water until spreadable.
Chocolate Topping: Melt chocolate and butter over hot water.
Spread cooled base with Peppermint Filling; refrigerate until set. Spread with Chocolate Topping; refrigerate until set before cutting.