Place pizza stone or steel on the middle rack and heat the oven to 475°F. Let the stone heat for at least 30 minutes after the oven reaches temperature, and take the dough out of the fridge to come to room temperature.
Cut off both ends of the squash, peel it, halve it lengthwise, and scoop out the seeds. Cut the flesh into 1/2-inch cubes, keeping them as even as possible, and measure out 3 cups.
Toss the cubes with 2 tablespoons olive oil, the maple syrup, 1/2 teaspoon salt, 1/4 teaspoon pepper, and the cinnamon or nutmeg. Spread in a single layer on a rimmed baking sheet and roast at 475°F for 20 to 25 minutes, stirring once, until deeply golden and a fork slides in easily.
Heat the remaining 1 tablespoon olive oil in a cast iron skillet over medium heat. Cook the sliced onion with a pinch of salt for 6 to 8 minutes until soft and just starting to color, then add the garlic and cook 30 seconds more. Scrape into a bowl, then fry a few extra sage leaves in the hot skillet for 20 to 30 seconds per side until crisp and drain on paper towels.
Dust your work surface with semolina or cornmeal and place the dough in the center. Press outward from the middle with your fingertips, leaving a thicker border, then lift and stretch the dough by gravity until you have a 14-inch round. Transfer to a piece of parchment dusted with more semolina.
Brush the dough lightly with olive oil all the way to the edges. Scatter the mozzarella evenly over the dough leaving a 1-inch border, then distribute the onion mixture, roasted squash, crumbled goat cheese, 12 fresh sage leaves, grated Parmigiano Reggiano, and the remaining salt and pepper.
Slide the pizza, parchment and all, onto the hot stone using a pizza peel or the back of a baking sheet. Bake for 10 to 13 minutes until the crust is golden brown, the cheese is bubbling, and the bottom feels firm. If the top needs more color, move the pizza to the upper rack for the last 60 to 90 seconds.
Let the pizza rest for 3 minutes so the cheese sets, then scatter the baby arugula over the hot surface. Add torn prosciutto if using, drizzle the hot honey in thin ribbons, top with fried sage leaves and a pinch of red pepper flakes, slice into 8 pieces, and serve right away.