Heat 2 tablespoons of the olive oil in a large saute pan over medium heat until hot. Add the onions, season with salt so the onions release their liquid, and reduce the heat to medium-low. Cook until the onions stop releasing water and the pan is nearly dry again, about 4 minutes. Do not allow the onions to brown. Add the water, stir, cover, and cook until softened, about 10 minutes. Uncover, stir in the thyme, and cook over medium heat until the onions are light brown, about 10 minutes longer. Season lightly with pepper. Do not cook the onions too long now as they will brown further in the oven. Let cool for several minutes.

Preheat the oven to 500 degrees F. Dust 2 baking sheets with cornmeal. Cut the dough into 3 equal pieces. Working on a floured surface, shape each piece of dough by hand or with a rolling pin into an 8-inch round, keeping the edges a little thicker than the center. Transfer the rounds to the prepared baking sheets. Brush each round with 1 teaspoon of the olive oil and divide the onions evenly among them. Let rise for about 20 minutes.

Push the dough with your fingers. Top each round with the caramelized onions and then scatter olives on top. Dot rounds evenly with the cheese. Bake until light brown all around the edges and on the bottom, about 12 minutes. Let cool for a minute, then, if desired, pile a handful of parley leaves on top of each. If using, put the vinegar in a spray bottle and mist the parsley.

Chef's Note: Diced cooked potatoes are also a good addition to the onions.