Start by mixing together water, sugar, salt, flour and yeast, until dough is formed. I use my hands to do so, but you can use a spoon. This is a high hydration dough: it will feel sticky in the beginning.
When there are no more dry visible spots and dough is formed, let it rest for 30 minutes, covered with a damp tower or plastic wrap.
After 30 minutes, uncover dough and pour around the edges 1 tablespoon of olive oil. That will help with the stretches and folds.
Start to stretch and fold the dough. Pull up on one side of the dough, fold it back down on itself. Repeat it 4 to 6 times through all the sides of the dough. Let the dough rest covered for another 20 minutes.
Repeat the stretch and fold 3 times more, giving it 20 minutes between each.
While the dough rests between stretches and folds, start caramelizing the onions. In a heavy-bottom pan, over medium heat, cook onions with butter, salt and pepper until it shrinks to half its size, around 10 minutes.
Add brown sugar and balsamic vinegar, and simmer for 20 minutes more. Reserve. It should be lukewarm to room temperature by the time the dough is ready.
After the last stretch and fold, let the dough rest for another 20 minutes before opening it.
Line a rectangular 9x13 inches baking sheet, and open the focaccia over it, using your hands. Cover the dough and let it rise while the oven preheats to 425 degrees Fahrenheit.
Distribute parmesan and caramelized onions evenly through the dough, topping it with the remaining 2 tablespoons of olive oil.
Using your fingertips, poke dimples all over the dough.
Bake focaccia for 20 to 25 minutes, or until bread is golden brown at the top and bottom.