Make the leftover dough by mixing 1 cup of the flour, 1/2 cup of the water, and the yeast in a bowl. Add 1/4 teaspoon of the salt and mix it in. Let this mixture triple in volume and punch it down.
Put the remaining 2 cups flour, 1/2 cup water, and 1/2 teaspoon salt in the bowl of a stand mixer fitted with the dough hook. Add 1/4 cup of the olive oil and the leftover dough. Mix on slow speed for 2 minutes. Cover the dough with a moist towel or plastic wrap and let rest for 20 minutes. Knead the dough for about 7 minutes, or until the dough is smooth and passes the windowpane test. Turn the mixer to high speed if needed to get the dough to slap against the sides of the bowl. Cover the dough with plastic wrap and let rise at room temperature for about 2 hours, or until doubled in volume.
Heat the remaining 3 tablespoons olive oil in a heavy-bottomed pan over medium heat. Add the onions and cook for about 30 minutes, stirring frequently, until the onions soften and turn pale brown. Cover with plastic wrap (to prevent them from turning gray).
Brush a 13 by 17-inch sheet pan with olive oil. Depending on the stickiness of the dough, you can either roll it or press it into shape. You may have to work in 2 or 3 stages, letting the dough rest for 10 minutes in between, to get it into the rectangular shape of the sheet pan. Press along the edges of the dough to form a border and press on the dough with your fingers to make dimples. Cover with plastic wrap and let rest at room temperature for about 1 hour, or until at least twice as thick.
Preheat the oven to 450°F. Spread the onions and olives over the dough, leaving a 1-inch border, drizzle with olive oil, and sprinkle with coarse salt. Don't press on the dough. Bake for about 25 minutes, or until the edges of the focaccia are golden brown. If the focaccia browns unevenly, turn the pan around halfway through the baking. Let cool before serving.