In a skillet, heat the olive oil over medium heat. Add the onion and a pinch of salt and cook, stirring often, until browned and caramelized, 30 to 40 minutes. Stir in the sage and cook for 2 more minutes.
Meanwhile, put a pizza stone in the lowest position in the oven and preheat the oven to 500 degrees F. (If you don't have a pizza stone, you can use a baking sheet.)
On a piece of parchment on a pizza peel or the back of a baking sheet, flatten out half the dough into a 10-inch round. Top with half of the ricotta, half of the Parmesan, a good pinch of salt and few turns of pepper, half of the onion and half of the squash shavings. Brush the squash with a thin even layer of olive oil and sprinkle with salt and pepper.
Slide the pizza onto the pizza stone (or heated baking sheet) and bake until the crust and squash are lightly browned, about 8 minutes. Repeat with the rest of the toppings on the other half of the dough.
While the pizza is baking, toss the arugula with the lemon juice, a pinch of salt and a few turns of pepper.
Top the pizzas with the arugula salad, additional Parmesan shavings and a sprinkle of crushed red pepper.