Stir the poolish into bowl of a stand mixer and add 2 cups of lukewarm water, 2 teaspoons yeast, 2 cups bread flour, 2 cups all purpose flour, and salt. Mix using dough hook. A smooth batter will form, then add remaining 1 cup all purpose flour and knead in stand mixer for 10 minutes.
This is a sticky dough and it will stick to bottom of bowl. Use a scraper to transfer dough to a lightly floured work surface and use your floured hands to knead dough, adding more bread flour as needed to prevent sticking, until dough is smooth and elastic, about 4 minutes (See note 1). Transfer the dough to a large, lightly greased bowl, cover the bowl with plastic wrap, and let the dough rise until it's nearly doubled in size, about 1 hour.
On a work surface sprinkled with flour, turn the dough out. This is a very wet, sticky dough. Fold the dough over a couple of times, using enough flour to keep hands dry and form dough into a round. Next, cut the dough in half, then each half into 4 or 8 equal pieces, depending on how large you want your barbari bread loaves to be (See Note 2).
Shape each piece of dough into a ball by rolling on surface under your palm, or use both hands. Transfer dough balls to a baking sheet, keeping them 2 inches apart. Tent the balls with lightly greased plastic wrap, and allow them to rest for 30 minutes.
Preheat the oven to 500°F. If you have a pizza stone, set it on the lowest rack or oven floor. If you'll be using an upside down sheet pan, position on rack in the center of the oven.