In a large mixing bowl or the bowl of the stand mixer, combine 5½ cups of flour(s), yeast, sugar, salt, olive oil, pizza dough flavor (is using) and the water.
Mix and knead until a smooth and elastic dough forms, about 7~9 minutes.
Place the dough in a lightly greased bowl, cover and let it rise for 1~2 hours, depending on your schedule.
In the meantime, get the toppings ready.
Caramelize the Onions: Heat 1tbsp oil in a nonstick pan and add the onions and salt; Cook on medium flame, stirring frequently, until nicely caramelized. Season with salt & pepper. Keep ready.
Cook the eggplant: In another large nonstick pan, shallow fry the eggplant slices until golden brown on both sides. Season with salt and pepper. Remove from the pan and keep ready.
Preheat oven to 450°F.
Gently deflate the dough and divide it into 4 pieces, for medium crust pizzas or 3 pieces for thicker pieces. Roll each piece, on a floured surface, into a 12" circle. If the dough doesn't stretch that easily, then let the dough rest to relax it and make it more cooperative.
Place the rolled pizzas on baking sheets or on a pizza stone. Cover and set aside for 15~20 minutes.
Brush the crust with little olive oil; spread pizza sauce over the surface and top with grated cheese and your favorite toppings. For the caramelized onions pizza, pizza sauce is usually not added, but it's your choice.
For the eggplant pizza, sprinkle the cheese on the base, arrange the eggplant slices and cherry tomatoes on top.
Bake the pizzas for 12~16 minutes, until they're golden, the toppings are hot and bubbly and the cheese is melted. Remove from the oven and transfer to a cooling rack. Cool for about 5~10 minutes, slice and serve.