Heat oven to 425° F. Shape the dough into a large oval and place on a cornmeal-dusted baking sheet.
In a large bowl, toss the squash, onion, pine nuts, thyme, 1 tablespoon of the oil, ½ teaspoon salt, and ¼ teaspoon pepper.
Scatter over the dough and sprinkle with the cheese. Bake until golden brown and crisp, 20 to 25 minutes.
Toss the arugula with the remaining tablespoon of oil and ¼ teaspoon each salt and pepper. Serve with the flat bread.