Combine the flour, yeast, and salt in the bowl of a standing mixer fitted with a dough hook. Add the water and oil and give the mixture a few turns with a rubber spatula until a rough dough forms. Knead on medium speed until the dough comes together but is still a little ragged, about 7 minutes. Transfer the dough to a lightly oiled bowl or dough-rising container, cover the bowl with plastic wrap, and refrigerate overnight. It will double in volume.
One hour before you want to bake, remove the dough from the refrigerator. Turn it onto a lightly floured countertop and use a bench scraper or sharp chef’s knife to divide it into 2 equal pieces for large flatbreads or 6 equal pieces for small ones. Shape the dough pieces into loose rounds, sprinkle with flour, and drape with plastic wrap. Let stand until puffy and almost doubled in volume, 1 hour.
While the dough rounds are proofing, place a baking stone on the middle rack of the oven. Preheat the oven to 400°F.
Line a baker’s peel with parchment paper. Use your hands to stretch the larger dough pieces into 8-inch circles or press the smaller ones into 3-inch circles. Place the dough circles on the parchment-lined peel. Press your fingertips into the dough rounds to create little dimples. Brush the rounds liberally with olive oil and sprinkle with za’atar.
Slide the dough, still on the parchment, onto the hot baking stone and bake until lightly browned, about 10 minutes for large breads, 7 minutes for small ones (don’t overbake or they’ll be too hard). Slide the breads, still on the parchment, onto a wire rack to cool slightly. Serve warm, or cool completely and serve at room temperature.
Mana’eesh are best eaten on the day they are baked. For longer storage, freeze in a zipper-lock plastic bag for up to 1 month. To defrost, place on the countertop for 15 to 30 minutes, and reheat in the oven at 350°F for 5 minutes before serving.