Heat oven to 425° F. Shape the dough into a large oval and place on a cornmeal-dusted baking sheet.
In a medium bowl, toss the sweet potato, shallots, thyme, 3 tablespoons of the oil, ½ teaspoon salt, and ¼ teaspoon pepper.
Scatter the potato mixture over the dough and top with the Brie. Bake until golden brown and crisp, 20 to 25 minutes.
Meanwhile, in a bowl, whisk together the vinegar, remaining tablespoon of oil, and ¼ teaspoon each salt and pepper.
Toss the dressing with the greens. Serve with the flat bread.