Heat oven to 350^.
Spray a 15x10-inch jelly roll pan with flour non stick cooking spray.
Beat eggs and vanilla in a large bowl with mixer for 4 minutes or until light and fluffy. Gradually beat in sugar. With mixer on low speed, beat in w/3 c. of water. Add flour and baking powder. Beat at low just until combined. Spread batter evenly in prepared pan.
Bake for 15-20 minutes until cake springs back when lightly touched. Immediately loosen edges from the pan. Turn out onto towel sprinkled with powdered sugar. ( I always cut the crust off the sides of the cake) ,then roll up towel and cake together, jelly-roll style, starting from a long side. Cool on wire rack. Unroll cake and remove towel.
Stir ice cream just until softened. Spread over unrolled cake to within 1/2-inch of edges. Reroll. Diagonally cut 4-inch piece from an end. Place long roll on serving plate. Place cut edge of piece against side to resemble branch. Place in freezer while preparing frosting.
Combine sweetened condensed milk and chocolate in heavy saucepan. Cook and stir over medium heat until chocolate melts and mixture thickens, about 6 to 8 minutes. Remove from heat. Stir in 3 Tablespoons water and peppermint extract. Cook, stirring rapidly, until mixture thickens again, 3 to 4 minutes. Cool 15 minutes or until mixture is thick enough to spread.
Remove chilled cake from freezer and frost quickly. Return to freezer for 15 minutes. With fork, score wavy lines in frosting . Freeze at least 2 hours. Remove from freezer 5 minutes before serving.