Divide dough in half. Color one half with green food coloring, knead until desired shade, place in plastic wrap and roll into a 2-inch cylinder; freeze until cold, about 1 1/2 hours. Knead other half with cocoa, flatten into a disc, wrap in plastic and chill thoroughly.
Unwrap green dough and slice into 1/8 -inch-thick rounds. Space well apart on 2 cookie sheets.
Roll chocolate dough between plastic wrap to 1/8 -inch thickness. Using a knife, cut triangles for hats (freeze for a few minutes if dough is too soft). Force chocolate scraps through a garlic press to make hair; place strands on sides of each green face. Put a triangle on top for hat. Freeze until very chilled, 15 minutes.
Preheat oven to 350F.
Bake cookies right out of freezer, rotating sheets halfway through baking, until done, about 10 minutes. Cool on sheets. Pipe on brown icing eyes. Dab on an icing dot and stick candy to it for a nose.
