Preheat the oven to 400 degrees F.
For the sauce: Over medium heat in a saucepan, bring cauliflower, water to cover, salt and white pepper to a simmer. Reduce the heat and allow to cook until soft, for 7 to 8 minutes. Remove from the heat, drain and puree until smooth. After pureeing, add the Parmesan and heavy cream as needed and puree again.
For the vegetables: In a saute pan over high heat, add the oil and allow to heat until on the verge of smoking. Then reduce the heat to medium and add the squash, zucchini, salt and black pepper. Cook, stirring until softened, 3 to 4 minutes. Remove from the heat and add the thyme.
For the pizza: Place the flatbreads on a cookie sheet and spread the sauce on top, evenly coating from edge to edge. Add half the zucchini, squash, corn, broccoli and top with shredded Parmesan. Once built, repeat the process with the second pizza. Bake until crisp, 7 to 8 minutes. Once cooked, remove from the oven and top with the green onions. Slice and serve.