Preheat the oven to 400°F. If you're using a baking stone/steel to bake your pizza, place the stone/steel in the bottom third of the oven before you start preheating).
Place the squash slices on a large rimmed baking sheet. Drizzle with the tablespoon of olive oil, season generously with salt and pepper, and toss well to coat with the oil. Roast the squash for 25 minutes, then flip the slices and roast for another 10 minutes or until the squash is caramelized and very tender. Remove the baking sheet from the oven and set it aside to cool.
Increase the oven temperature to 500°F (if using a baking sheet to bake your pizza) or the highest temperature your oven will allow (if using a baking stone/steel). If you're using a baking stone or steel, once the oven has reached its highest temperature, switch the oven setting to Broil on high and let it heat up for about 15 minutes.
Stretch or roll out your dough to a 12- to 14-inch circle, then transfer it to an oiled or parchment-lined baking sheet or a lightly floured pizza peel (if using a baking stone/steel).
Brush the dough lightly with olive oil, making sure to get all the way to the edges. First arrange the apples in a pretty pattern, then add the squash. Sprinkle on the cheese and season the pizza with salt and pepper.
Transfer the pizza to the oven and bake until the crust is golden and the cheese has browned in spots—8 to 10 minutes on the baking sheet, 6 to 8 minutes on the baking stone/steel.
Remove the pizza from the oven and let it rest for 5 minutes, then sprinkle on the arugula and give the pizza a light drizzle of olive oil and a pinch of salt; slice and serve.