Mix white sugar and cinnamon together in a small bowl.
Roll out 1 thawed loaf into a large rectangle. Brush half of the melted butter evenly over dough. Sprinkle half of the cinnamon sugar mixture over butter. Roll up dough into a log starting with the long edge closest to you. Cut log into 1-inch slices with a sharp serrated knife. Arrange slices cut-side up in an ungreased 9x13-inch baking pan.
Repeat step 2 with second thawed loaf.
Combine brown sugar, vanilla ice cream, and 1/2 cup butter in a small saucepan over medium heat; stir until melted and well-blended, 3 to 5 minutes. Pour caramel mixture over rolls in the baking pan. Cover with aluminum foil and refrigerate until rolls double, 8 hours to overnight.
Preheat oven to 350 degrees F (175 degrees C).
Bake in the preheated oven until golden brown, about 30 minutes. Allow to cool in the pan for 10 minutes before serving.