Sprinkle yeast over warm water in a small bowl. The water should be no more than 100 degrees F (40 degrees C). Let stand for 5 minutes until yeast softens and begins to form a creamy foam.
Sift flour, sugar, 1 tablespoon oregano, and 1 teaspoon salt into a large bowl. Mix egg and oil into dry ingredients; stir in yeast mixture. When dough has pulled together, turn out onto a lightly floured surface and knead until smooth and elastic, about 8 minutes. Lightly oil a large bowl, then place dough in bowl and turn to coat with oil. Cover with a light cloth and let rise in a warm place (80 to 95 degrees F (27 to 35 degrees C)) until doubled in volume, about 1 hour.
Preheat oven to 450 degrees F (235 degrees C). Lightly grease a baking sheet.
Cook and stir diced tomatoes, tomato paste, onion, 1 tablespoon oregano, garlic, onion powder, 1 teaspoon salt, and pepper in a saucepan over medium heat. Cook until sauce has thickened, 15 to 20 minutes.
Punch down dough and turn it out onto a lightly floured surface. Use a knife to divide dough into two equal pieces--do not tear. Shape dough into rounds and let rest for 10 minutes. Form dough into rectangles, and place onto prepared baking sheet. Spoon tomato sauce over dough. Sprinkle with 1 1/4 cup mozzarella cheese; top with bell pepper, onion, and mushroom. Sprinkle 1/4 cup mozzarella cheese over top.
Bake in preheated oven until crust is browned and crisp and cheese has melted, 20 to 25 minutes.