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To make the manakeesh dough: In the bowl of a stand mixer, add the flour, salt, yeast, and sugar, and stir to combine with a wooden spoon or the dough hook (not attached), mixing by hand. While mixing, gradually add the water, then continue to mix by hand until a shaggy dough forms, about 1 minute.
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Attach the dough hook, set the mixer to medium-high speed, and knead the dough until smooth and slightly sticky, about 3 minutes.
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Coat a medium mixing bowl with the oil. Gather the dough into a ball, transfer it to the prepared bowl, and flip it to coat it completely with the oil. Tightly cover the bowl and let the dough rise until it’s almost doubled in size, about 45 minutes.
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Transfer the dough to a clean work surface and divide it into four pieces (about 108g each). Shape each piece into a ball, then place the balls about 3" apart on the work surface or on a baking sheet, cover, and let rise until almost doubled in size, about 45 more minutes.
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While the dough is rising, preheat the oven to 550°F or its highest temperature with a baking steel, baking stone, or overturned dark, heavy baking sheet on a rack in the lowest position (remove any other racks to get them out of the way).
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To make the za’atar manakeesh topping: In a small bowl, combine the za’atar, olive oil, and salt.
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To make the cheese manakeesh topping: In a medium bowl, combine the garlic, onion, cheese, olive oil, and dried mint. Finely chop the fresh mint and set aside for garnishing.
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Lightly dust a baker’s peel with flour and set aside near your work area.
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Working with one piece at a time, place a ball of dough on a lightly floured surface and dust the top with flour. Press to flatten the dough, then roll it into a 9" round. As you roll, stop to lift the dough and rotate it to create an even round and prevent sticking.
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Dock the dough with a fork or docker, pricking the top all over.
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Transfer the dough to the prepared peel. Stir the za’atar mixture to re-combine, then spoon a quarter of the mixture (about 1 1/2 tablespoons or 22g) onto the center of the dough. Use the back of a spoon or an offset spatula to spread the mixture over the dough evenly, leaving an 1/8" border bare.
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Shake the peel to ensure no spots are sticking (if spots stick, lift the dough and dust more flour underneath), then transfer the dough from the peel to the pre-heated stone, steel, or baking sheet in the oven.
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Bake for 3 to 4 minutes, until the edges turn golden brown and slightly char in spots. If needed, use heatproof tongs to rotate the bread for even browning. If using the cheese topping, add about a quarter of the cheese mixture (a heaping 1/4 cup or 67g) to the bread 2 minutes into baking, then return the man’oushe to the oven for 1 to 2 more minutes.
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Remove the bread from the oven using tongs or the peel, then transfer to a cutting board to cool for at least 3 minutes. Garnish with fresh mint, if desired.
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Continue shaping, topping, and baking the remaining manakeesh, making sure your peel is cool before placing the dough on top.
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To serve, quarter each man’oushe with a pizza wheel or chef’s knife, or fold it in half and wrap it in wax or parchment to eat on the go.
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Storage information: Store leftover manakeesh in the refrigerator for up to 2 days, or freeze for up to 3 months. Reheat in a 250°F oven for about 10 minutes (if frozen, thaw before reheating).