In a large bowl combine shortening and butter. Beat with an electric mixer on medium to high speed for 30 seconds. Add sugar, baking powder, and salt. Beat until combined, scraping sides of bowl occasionally. Beat in eggs, milk, and vanilla until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour.
Divide dough into three portions. Leave one dough portion plain. Knead red food coloring into another dough portion. Knead green food coloring into the remaining dough portion. If necessary, cover and chill dough for 1 hour or until easy to handle. Wrap each dough portion in plastic wrap or waxed paper. Chill for 1 to 2 hours or until dough is easy to handle.
Divide the green dough portion into thirds. Shape each portion into a 11x1-inch log. Wrap with plastic wrap and freeze about 30 minutes or until firm.
Meanwhile, divide the plain dough portion into thirds. Place each portion between two sheets of waxed paper; roll each portion into an 11x4-inch rectangle. Freeze about 15 minutes or until firm.
Divide the red dough portion into thirds. Place each portion between two sheets of waxed paper; roll each portion into 11x5-inch rectangle. Freeze about 15 minutes or until firm. (If necessary, dough rectangles can be stacked on top of each other in the freezer.)
To assemble, remove top sheet of waxed paper from each portion of plain dough. Place each green dough log on a long edge of a plain dough rectangle. Roll up plain dough around the green log to completely encase the green dough. Trim excess plain dough if necessary. Pinch dough to seal edges. Place each of these plain-and-green logs on a red dough rectangle. Roll up red dough around log to completely encase the plain dough. Wrap logs in plastic wrap or waxed paper. Freeze about 1 hour or until firm.
Preheat oven to 375 degrees F. Grease cookie sheet or line sheet with parchment paper. Cut logs into 1/4-inch slices. Place slices 2 inches apart on the prepared cookie sheet.
Bake in the preheated oven for 8 to 10 minutes or until bottoms are light brown. Cool on cookie sheet for 1 minute. Transfer to a wire rack; cool completely.
If desired, place frosting in a pastry bag fitted with a small round tip (or place frosting in a resealable plastic bag and snip a small hole in one corner of bag). Pipe four small dots onto each cookie to make it resemble a button.