Place a baking sheet in the upper third of the oven and an inverted baking sheet (or pizza stone) in the lower third; preheat to 475 degrees F. Toss the cauliflower with 2 tablespoons olive oil, the red pepper flakes and 1/4 teaspoon salt in a medium bowl. Spread in a single layer on the top hot baking sheet. Roast until the cauliflower is golden brown on the bottom, 10 to 12 minutes.
Meanwhile, combine 1 tablespoon olive oil and the garlic in a medium saucepan over medium-high heat. Cook, stirring frequently, until the garlic just starts to turn golden, about 2 minutes. Add the tomatoes and 1/2 teaspoon salt and simmer until reduced to about 1 cup, about 8 minutes; let cool. Combine the parsley, olives, capers, lemon zest and juice, and 2 tablespoons olive oil in a medium bowl; set aside.
Brush a large piece of parchment paper with the remaining 2 tablespoons olive oil. Stretch the pizza dough into an 11-by-15-inch rectangle on the parchment. Transfer the dough (on the parchment) to a pizza peel or another inverted baking sheet. Spread the tomato sauce all over the dough, leaving a 1/2-inch border. Top with the mozzarella, then scatter the roasted cauliflower on top and sprinkle with the parmesan. Slide the pizza (on the parchment) onto the bottom hot baking sheet. Bake until the crust is golden, 20 to 25 minutes. Top with the parsley-olive mixture; add anchovies, if desired. Sprinkle with red pepper flakes.
Let your pizza dough come to room temperature (about 30 minutes) before stretching it. If the dough is still difficult to work with, let it sit for a few more minutes.