Mix together the warm water and sugar. Sprinkle yeast on top and let it stand, covered with plastic wrap in a warm place, for about 10 minutes or until foamy. If it does not get foamy, the water may not have been hot enough or the yeast may no longer be active. This is called proofing the yeast.
Meanwhile, combine the flour and salt in a large bowl. Add the olive oil and work it in with your fingers. Make a well in the center, add the yeast and water mixture, and stir to form a soft dough. Turn the dough out onto a lightly floured surface and knead for about 10 minutes until smooth and elastic. Form it into a ball and place it in a lightly oiled bowl.
Cover the bowl with a damp cloth or plastic wrap and leave the dough to rise in a warm place for about 1 hour until it doubles in size.
Preheat oven to 400°F. Place a baking stone or baking sheet in the oven as it heats. After the dough has risen for an hour, knead it briefly and divide it into 6 or 8 balls. Place the balls on a lightly oiled baking sheet, cover with a damp cloth or plastic wrap, and let stand for about 15 minutes. In a small bowl, combine the za’atar and the oil, stirring well. Set aside. On a lightly floured surface, flatten each ball of dough and roll it into a circle 1/8-inch thick and about 7-8 inches in diameter. Press each circle with your fingertips to make little indentations for the topping to rest in. Spread 1 heaping tablespoon of za’atar topping over each round, leaving a 1/2-inch border around the edges.
Preheat oven to 400°F. Place a baking stone or baking sheet in the oven as it heats. After the dough has risen for an hour, knead it briefly and divide it into 6 or 8 balls. Place the balls on a lightly oiled baking sheet, cover with a damp cloth or plastic wrap, and let stand for about 15 minutes. In a small bowl, combine the za’atar and the oil, stirring well. Set aside. On a lightly floured surface, flatten each ball of dough and roll it into a circle 1/8-inch thick and about 7-8 inches in diameter. Press each circle with your fingertips to make little indentations for the topping to rest in. Spread 1 heaping tablespoon of za’atar topping over each round, leaving a 1/2-inch border around the edges. Using a large spatula or a pizza peele, slide the rounds, one or two at a time, on the heated pizza stone or baking tray. You can use a wire mesh pizza pan and assemble the bread right on top and then place the whole thing in the oven. Bake until lightly browned and crisp, about 8 minutes. Depending on the size of your oven and baking stone or baking sheet(s), you may need to do several batches. Serve warm right away or else cool on racks.
