In a small bowl, combine warm water, sugar, and yeast; allow to bloom for 10 minutes until frothy.
In a large mixing bowl or stand mixer, combine flour and salt, then add the yeast mixture and 1/4 cup olive oil.
Knead the dough for approximately 8 minutes until smooth and elastic; place in a greased bowl, cover, and let rise for 1 hour until doubled in size.
Preheat oven to 450°F (230°C) and grease two large baking sheets.
Punch down the risen dough and divide into 8 equal balls; roll each ball into a circle approximately 6 inches in diameter and 1/4 inch thick.
Use your fingertips to create small indentations across the surface of each dough circle to prevent air bubbles and hold the toppings.
In a small bowl, whisk together the za'atar spice blend and 1/3 cup extra virgin olive oil.
Spread the za'atar mixture onto 4 dough rounds and distribute the shredded cheese evenly across the remaining 4 rounds.
Transfer the dough rounds to the prepared baking sheets and bake for 10 to 12 minutes until the crust is golden brown and the cheese is melted.
Remove from the oven and serve warm, optionally folding the flatbreads while eating.