Remove dough from canisters, fold each roll of dough in half and wrap together loosely with plastic wrap. Place wrapped dough in the refrigerator and let rest at least 2 hours.
Preheat oven to 350 degrees F. Lightly grease 2 (8-inch) cake pans.
In a small bowl, combine brown sugar, cinnamon, and instant coffee. Set aside. On a lightly floured surface, place 1 piece of the folded dough on top of the other piece of dough and flatten. Roll dough into an 8 by 14-inch rectangle. Spread softened butter over surface of dough and sprinkle with the sugar-cinnamon-coffee mixture. Starting with the long edge, roll dough like a jelly roll. Cut into 12 pieces and divide between the 2 prepared cake pans. Bake 25 to 30 minutes until golden brown. Allow to cool 10 minutes before spreading with icing.
For icing: In a medium mixing bowl with a handheld mixer, beat together softened butter and sifted powdered sugar adding the sugar 1 cup at a time. Beat until light and fluffy. Add remaining ingredients and beat until combined. Spread or drizzle over cooled rolls.