Preheat the oven to 375°F and spray a 9x13 baking dish with nonstick cooking spray.
Remove the cinnamon rolls from the package and set the icing aside. Cut each cinnamon roll into six pieces and place the pieces in an even layer in a baking dish.
In a large bowl, whisk together the pumpkin puree, evaporated milk, brown sugar, eggs, and pumpkin spice. Stir until smooth.
Pour the pumpkin mixture over the top of the cinnamon rolls.
Cover the pan with foil and bake in the preheated oven for 25 minutes, then remove the foil and bake for an additional 10 to 15 minutes, until the cinnamon rolls are golden brown.
Allow the casserole to rest for 5 to 10 minutes, then drizzle the cinnamon roll icing over the top before serving.