In a large bowl or a stand mixer fitted with the hook attachment, mix the flour, water, yeast, and salt until it comes together in a shaggy mixture, about 1 minute. Add the olive oil and, using your hands right in the bowl or with the dough hook, thoroughly knead the dough until it transforms from a tight, rough mass into a smooth, springy, cohesive ball, about 5 more minutes.

Cover the dough in the bowl with a slightly damp kitchen towel or a piece of oiled plastic wrap. Set the bowl aside at room temperature until the dough has puffed and risen to double its volume, about 60 minutes.

Preheat the oven to 450°F and line a large baking sheet with parchment paper.

Lightly flour a flat surface, transfer the dough ball from the bowl to the surface. Divide the dough into 12 equal pieces, and using your palms, shape them into round balls. Set aside to rest on the prepared baking sheet covered with a slightly damp kitchen towel for 10 minutes to relax the dough’s elasticity.

Take a ball of dough and press it into a round, flat, palm-sized disc about ¼-inch thick and 4 inches in diameter. Return the flatbread to the prepared baking sheet and cover again with the kitchen towel while you work on the rest. Use a fork to poke each flatbread to prevent it from puffing up too much in the oven.

Sprinkle the flatbreads with about 1 tablespoon of mozzarella cheese and 1 tablespoon of Akawi cheese. Bake until they are slightly puffed, golden brown on the bottom and the cheese has melted, 10 to 13 minutes. Let the manakeesh cool for at least 5 minutes before serving.