Activate the yeast.
In a large bowl, combine the warm water, sugar, and yeast. Let stand for 5–10 minutes, until foamy.
Make the dough.
Add the flour and salt. Mix by hand or with a stand mixer fitted with a dough hook, gradually adding 1 tablespoon olive oil.
Knead for 5–7 minutes, until smooth and elastic. The dough should be soft and slightly tacky.
Let rise.
Lightly grease the bowl and place the dough inside. Cover and let rise in a warm place until doubled in size.
Begin checking after 15–20 minutes in a very warm kitchen, but allow longer as needed. The dough’s size is a better guide than the clock.
Divide and rest.
Gently press down the dough and divide into equal-sized balls. Arrange on a tray, cover, and let rest for 5–10 minutes.
Prepare the zaatar topping.
Stir together the zaatar, vegetable oil, and olive oil until evenly combined.
Shape and top.
Lightly oil your hands and flatten each dough ball into a round disk. Place on baking trays lined with parchment suitable for 450°F, or use lightly greased trays.
For zaatar manakeesh, spread a thin, even layer of the topping over each round, leaving a small border.
Rest and preheat.
Let the rounds rest for another 5–10 minutes. Meanwhile, preheat the oven to 450°F.
Bake the zaatar manakeesh.
Bake for 10–15 minutes, until the edges and undersides are lightly golden and the bread is cooked through.
Bake the cheese variation.
For cheese manakeesh, bake the untopped rounds for about 7–8 minutes first. Add the shredded cheese, then return to the oven until melted and lightly golden and the bread is fully baked.
Serve warm.
Serve with labneh, olives, fresh vegetables, or your favorite sides.