Heat 2 tablespoons of olive oil in a large saute pan over medium heat.

Add the thinly sliced onions to the pan.

Season the onions with salt.

Reduce the heat to medium-low.

Cook the onions until they stop releasing water and the pan is nearly dry, about 4 minutes.

Ensure the onions do not brown.

Add the water to the pan and stir.

Cover the pan and cook until the onions are softened, about 10 minutes.

Uncover the pan, stir in the chopped fresh thyme.

Cook over medium heat until the onions are light brown, about 10 minutes.

Season lightly with pepper.

Do not cook the onions too long as they will brown further in the oven.

Let the caramelized onions cool for several minutes.

Preheat the oven to 500 degrees F (260 degrees C).

Dust 2 baking sheets with cornmeal.

Cut the pizza dough into 3 equal pieces.

On a floured surface, shape each piece of dough into an 8-inch round, keeping the edges a little thicker than the center.

Transfer the dough rounds to the prepared baking sheets.

Brush each round with 1 teaspoon of olive oil.

Divide the caramelized onions evenly among the rounds.

Let the dough rise for about 20 minutes.

Push the dough with your fingers to create dimples.

Top each round with the caramelized onions and then scatter pitted oil-cured olives on top.

Dot the rounds evenly with grated fontina or Gorgonzola cheese.

Bake until the edges and bottom are light brown, about 12 minutes.

Let cool for a minute.

Pile a handful of fresh flat-leaf parsley leaves on top of each schiacciata, if desired.

If using, put the aged red-wine vinegar in a spray bottle and mist the parsley.