In medium bowl, stir cookie mix, softened butter, flour, and egg until dough forms.
On a floured surface, roll dough to about ¼-inch thickness.
Brush with melted butter.
In a small bowl, mix together sugar and cinnamon. Sprinkle this mixture evenly over dough.
Begin to roll up dough, starting at the long side. If the dough tears at all, do not worry. This dough is very forgiving. Just pinch it back together, it will still bake up nicely.
Cut dough log in half. Wrap in plastic wrap and pop dough in the freezer for about 20-30 minutes (until firm – not hard).
Heat oven to 375F degrees. Cover your baking sheet with parchment paper. Take dough out of freezer and begin cutting into ¾-inch slices. If dough tears a little as you cut it, just pinch it together a little with your fingers. Place slices about 2 inches apart on cookie sheet. Bake 10-12 minutes or until edges are lightly golden brown. Cool for about 2 minutes, remove from cookie sheet to cooling rack. Cool completely.
In a small bowl, mix powdered sugar and milk until smooth (adding a tad more milk if necessary). Drizzle over cooled cookies (making sure you have aluminum foil or waxed paper underneath cooling racks to catch excess icing).