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, then 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. Arrange slices cut-side up in an ungreased 9x13-inch baking pan.

Repeat step 2 with the second thawed loaf.

Combine brown sugar, vanilla ice cream, and 1/2 cup butter in a small saucepan over medium heat, stirring until melted and well-blended, about 3-5 minutes. Pour caramel mixture over rolls in the baking pan. Cover with aluminum foil and refrigerate until rolls double in size (8 hours to overnight).

Preheat oven to 350°F (175°C).