Position a rack in the lower third of the oven. Place a pizza stone or upside-down baking sheet on the rack and preheat to 500 degrees F. Cover another upside-down baking sheet with parchment paper and brush with 2 tablespoons olive oil. Put the pizza dough on the oiled paper and turn to coat. Cover with another piece of parchment and set aside.
Mix the cheeses, red pepper flakes, and salt and pepper to taste in a bowl. Combine the scallion greens and herbs in another bowl. Toss the broccolini and scallion whites on a baking sheet with the remaining 1 tablespoon olive oil, and salt and pepper to taste. Roast 15 minutes.
Uncover the dough; stretch into a 12-by-16-inch rectangle. Season with salt and pepper, then slide the dough and parchment onto the hot stone or baking sheet. Bake until golden, 6 to 8 minutes.
Slide the crust and parchment back onto the baking sheet and remove from the oven. Top with the cheese mixture, then the roasted vegetables and herb mixture. Return the pizza to the oven and bake until the cheese melts, about 3 more minutes. Drizzle with olive oil.