Coat a baking sheet and some plastic wrap with the cooking spray. Place the rolls on the baking sheet and cover with the plastic wrap. Let thaw and rise for 2 to 3 hours.
Meanwhile, make the filling: Place the spinach in a kitchen towel and wring dry. Put in a large bowl, separating any clumps. Add the artichokes, mozzarella, ricotta, Parmesan and heavy cream and stir until combined. Season with pepper.
Preheat the oven to 400 degrees F.
Roll out each dinner roll on a lightly floured surface into a 5-inch round. Spoon 1/4 cup of the filling on half of each dough round. Dampen the edges of the rounds with water, then fold the dough over the filling to enclose, pressing the edges to seal.
Arrange the calzones on 2 baking sheets lined with parchment paper. Bake until golden brown, 10 to 13 minutes.
Serve the calzones with the warmed marinara sauce for dipping, and a salad.