Beat butter with an electric mixer on medium to high speed in a large mixing bowl for 30 seconds, then add 1 cup sugar and baking powder and beat the mixture until fluffy. Beat egg, vanilla, and peppermint extract into the mixture, then beat in with the mixer as much of the flour as you possibly can, and stir in the remaining flour with a wooden spoon.
Divide dough into 3 equal sections and stir red food coloring into one section, stir green good coloring into another section, and leave the last section plain. Cover each section with plastic wrap or foil and allow the dough to chill about 1 hour or until the dough is easy to handle.
Divide dough into 4 equal sections and roll each section on a lightly floured surface until it is about a 1/2-inch-diameter rope. Take a red, green, and plain colored rope, place them side by side, and twist the 3 ropes together, repeating with each remaining rope and allowing the twisted ropes to chill for 30 minutes.
Cut ropes into 1/2-inch slices to make larger cookies or 1/4-inch slices to make smaller cookies. Roll each slice carefully into balls and blend the colors as little as possible, then place each ball on an ungreased cookie sheet about 2 inches apart. Take each ball of dough and flatten so that dough is 1/4-inch thick using a glass dipped in additional sugar.
For larger cookies, bake 8 to 10 minutes (6 to 8 minutes for smaller cookies) or until edges are set in a 375 degree F oven. Transfer cookies to a wire rack and let cool.