Combine warm water, sugar, and yeast in a large bowl. Stir and leave 5–10 minutes until foamy. If no foam forms, start over with fresh yeast.
Add olive oil and salt to the yeast mixture. Add flour one cup at a time, stirring between additions until a shaggy dough forms. Turn onto a lightly floured surface and knead for 8 minutes until smooth and soft. Place in a lightly oiled bowl, cover with plastic wrap, and leave in a warm spot for 1 to 1.5 hours until doubled.
Mix za’atar and olive oil with a pinch of salt until a spreadable paste forms. Add more oil if too dry, more za’atar if too wet. Taste and adjust — add sumac if the blend tastes flat.
Preheat oven to 475°F with a baking stone, steel, or upside-down heavy baking sheet inside. Divide dough into 6 equal portions. Roll each into a thin round about 7–8 inches in diameter. Spread 1–2 tablespoons of za’atar paste over each round, going almost to the edges.
Slide manakish onto the hot surface using parchment. Bake for 6–8 minutes until edges are golden and the topping is set and slightly darkened. Work in batches.
Serve immediately while warm. Fold in half or roll with fresh tomatoes, cucumber, mint, and labneh inside — or cut into wedges and serve with dips alongside.
