Preheat oven to 375 degrees. Brush a baking sheet lightly with a little of the olive oil. Spread the broccoli and onions in a single layer on the baking sheet. Drizzle with 1 TB of olive oil, and toss with your hands to coat.
Roast the vegetables in the oven for about 15 minutes while preparing the pizza dough.
For the crust, in a large bowl stir together half of the flour with all other dry ingredients. Stir in the water and combine well. Gradually work in the remaining flour. Add additional 1 - 2 tablespoons of water as needed.
Gather the dough into a ball and place in a greased bowl. Cover with a damp, warm towel. Let rest and rise for 15 minutes.
Remove the vegetables from the oven, transfer them to another container. On the same baking sheet, press the risen dough onto the pan. Spread it out to desired thickness.
Evenly spread the Alfredo sauce on top of the dough. Sprinkle with half of the mozzarella and arrange the roasted vegetables and tomatoes. Top with the remaining mozzarella and Parmesan cheese.
Bake for 20 minutes on center oven rack or until the bottom of the crust is golden and the cheese is melted through.