Heat the oven to 350°F. Butter a metal 9-by-13-inch pan and cut each cinnamon roll into six pieces with kitchen shears.
Scatter the dough in an even layer, leaving small gaps. Whisk the eggs, milk, vanilla, cinnamon and salt until smooth.
Pour the custard evenly over the dough without pressing the pieces into a solid mass.
Bake for 30 to 35 minutes, tenting loosely after 20 minutes if needed. Check that the custard reaches 160°F and a dough piece from the middle pulls apart into baked layers with no sticky raw streak.
Cool for 10 minutes, then drizzle the reserved icing over the surface and cut into eight portions.