If using active dry yeast: Pour the warm water (100-105° F/38-40°C) into a small bowl. Stir in the sugar and yeast. Set aside for 5-10 minutes, until frothy. If using instant dry yeast: Add the yeast and the sugar with the flour.
Whisk together the flour and salt in a large mixing bowl or the bowl of a stand mixer, add 5 tablespoons of olive oil and mix together briefly using a spoon or the dough hook.
Add the yeast and water mixture and begin to knead. The mixture should come together as a soft, stretchy dough, pulling away from the sides of the bowl. Add a bit more flour if mixture is too wet, or a bit more water if mixture seems dry or too firm. Knead for 5-10 minutes, until dough is smooth, soft and elastic.
Transfer the dough to a large, clean, oiled bowl. Cover with plastic wrap and let dough rise until doubled in size, about 1 hour.
While the dough is rising, peel, halve and slice the onion lengthwise into very thin strips. Submerge the sliced onion in a bowl of cold, salted water and soak for 30 minutes. Drain onions well and dry with paper towels.
Preheat oven to 230°C (450°F) with a rack in the middle.
Once it has risen, punch down the dough and shape into a smooth ball. Pour 3 tablespoons of olive oil into a large cast iron skillet or medium sized pizza pan with at least 1”/2.5cm sides. Place the ball of dough in the middle of the pan and press out gently with your fingers. Let dough relax for about 10 minutes.
Continue to press dough out into the pan, letting it relax for a few minutes each time as necessary, until dough covers the bottom of the pan. It should take 3 – 5 repetitions, depending on the size of the pan.
Sprinkle the onions over the top of the dough. Drizzle a tablespoon of olive oil over the onions, and sprinkle with the dried oregano, rubbing it between your fingertips while doing so to bring out the flavour.
Place the fugazza in the centre of the oven and bake for about 20 minutes, until the edges start to turn golden brown. If desired, remove fugazza from oven after 15 minutes and top with thin slices of mozzarella and sprinkle with grated Parmesan then return to oven and bake until the fugazza is golden brown and crispy around the edges. Brown the onions under the oven grill or broiler for the last 2 – 3 minutes of cooking, if desired.
Remove from the oven, allow to cool enough to handle and cut into wedges or squares to serve.