Combine yeast, sugar, and 1⁄2 cup water heated to 115° in a bowl. Let sit until foamy, about 10 minutes. Combine 1 tbsp. salt and 1 cup warm water in the bowl of a stand mixer fitted with a dough hook; mix until salt is dissolved. With the motor running, slowly add flour in 3 batches and yeast mixture in 2 batches. Add 2 tbsp. oil; mix until a soft, sticky dough forms. On a floured surface, knead dough until smooth, 5-10 minutes. Shape dough into a ball and place in a lightly greased bowl. Cover loosely with plastic wrap and let sit in a warm place until doubled in size, about 1 hour.
Heat oven to 350°. Punch dough down; knead for 5 minutes. Divide into 6 pieces and roll each piece into a ball; place on a parchment paper-lined baking sheet and cover loosely with plastic wrap. Let sit in a warm place until doubled in size, 20-25 minutes.
Working with 1 piece of dough at a time, dust dough heavily with flour and roll into a 7" circle, about 1⁄8" thick. Transfer to a parchment paper-lined baking sheet. Bake, flipping once, until puffed, 3-5 minutes. Transfer pita to a plate and cover with a kitchen towel to keep warm.
Mix remaining oil, za'atar, juice, zest, and salt in a bowl. Heat a cast-iron grill pan over medium-high heat. Working with one pita at a time, cook, flipping once, until puffed and charred in spots, 2-3 minutes. Transfer to a serving platter and brush generously with za'atar mixture.