In a large bowl, sift in the all-purpose flour, add the dry yeast, salt, and sugar, then mix well.
Pour in the lukewarm water and olive oil, then start kneading. Work it into a smooth, elastic dough, then cover and let it rise for 1 hour.
While the dough is rising, prepare the filling. Slice the red onion thinly, and grate the cheddar and mozzarella cheeses.
In a skillet, sauté the red onion in a little olive oil until slightly caramelized.
Divide the risen dough into two parts, then roll them out into circles on a floured surface.
Spread tomato sauce on one half of each dough circle, then place the caramelized red onion and grated cheeses on top.
Sprinkle the filling with oregano, basil, and a pinch of black pepper.
Fold the other half of the dough over the filling, then press the edges together to seal, ensuring the filling doesn't escape.
Brush the top with beaten egg, and prick a few holes with a fork to allow steam to escape during baking.
Bake in a preheated oven at 400°F (200°C) until golden brown, about 20 minutes.
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