Preheat the oven to 500°F for at least 30 minutes while you make the dough and/or prepare the fillings. Line a heavy duty rimmed baking sheet with parchment paper.
Heat the olive oil in a large skillet over medium-high heat. Add the onion and garlic and cook for 5 minutes, stirring frequently, until the onion is soft. Add half of the spinach, stir just until it wilts, then add the rest of the spinach and the artichoke hearts. Stir until the spinach is completely wilted and remove the pan from the heat. Season to taste with salt and pepper.
Dust a clean work surface with flour. If you're making one large calzone, roll out the pizza dough into a 12-inch round. If you're making four individual calzones, divide the dough into four equal-size portions, shape each one into a ball, and roll them out into 6-inch rounds.
To fill the calzone(s): If you're making one large calzone, scoop the spinach mixture onto the bottom half of the dough round, leaving a 1-inch border of dough, and sprinkle on the cheese. If you're making four individual calzones, layer the spinach and cheese in the center of the dough rounds, leaving at least a ½-inch border of dough all the way around.
Brush the naked edges of dough lightly with water and fold the calzone(s) in half to cover the filling. Press the edges together, then crimp them with your fingers to create a tight seal.
Transfer the filled calzones to the prepared baking sheet. Brush the top(s) with olive oil, cut small steam vents in the top(s) using a small, sharp knife, and place them in the oven. Bake for 15 to 20 minutes or until the calzone(s) are golden brown.
Remove the calzone(s) from the oven and let them rest for 5 to 10 minutes before slicing. (The large calzone should be cut into eight pieces; the individual calzones can simple be cut in half or served as is.) Serve the calzones with marinara sauce for dipping.