Preheat the oven to 400 degrees F. Oil a baking sheet with 1 tablespoon olive oil and set aside.
Heat the butter in a large skillet over medium heat. Add the onion and garlic and cook for 2 minutes. Add the broccoli and season to taste with salt and pepper. Cook until the broccoli has softened slightly and become bright green, about 1 minute. Remove from the heat and set aside.
Lightly flour a clean work surface and stretch or roll the dough into a large rectangle, about 11-by-13 inches. With the long side closest to you, spread the pesto on the dough, leaving about a 1-inch border on all sides. (You'll want to maintain the 1-inch border when adding the remaining toppings.) Top the pesto with the broccoli mixture and sprinkle over the Cheddar. Shingle the mozzarella slices over the Cheddar. Sprinkle on the basil leaves.
Fold the short sides in, then roll the dough into a log. Place the stromboli seam-side down on the prepared baking sheet. Using a sharp knife, make 5 to 6 diagonal slits on top of the dough. Brush the top with the remaining tablespoon olive oil and sprinkle with the grated Parmesan.
Bake until golden brown, about 20 minutes.
Allow to cool for about 5 minutes before slicing and serving. Serve the warmed marinara alongside and top each serving with some more fresh basil.