Grease a 9-by-13-inch baking dish. Cut each cinnamon roll into 6 pieces and spread them in the dish.

Whisk the eggs, milk, vanilla, cinnamon, maple syrup, and melted butter in a bowl. Pour the mixture evenly over the dough.

Cover tightly with foil and freeze for up to 2 months.

Thaw overnight in the refrigerator. Bake covered at 350°F for 25 minutes, then uncover and bake 10 to 15 minutes more.

Cool for 5 minutes, drizzle with the reserved icing, and cut into squares.