In a small bowl, combine warm water, active dry yeast, and sugar. Stir gently and let sit for 5-10 minutes until foamy.
In a large mixing bowl, combine flour and salt. Make a well in the center and pour in the yeast mixture and 2 tablespoons olive oil.
Mix with a wooden spoon or your hands until a dough forms. Transfer to a floured surface and knead for about 8-10 minutes until smooth and elastic.
Place the dough in a lightly oiled bowl, cover with a clean kitchen towel, and let it rise in a warm place for about 1 hour or until doubled in size.
Preheat your oven to 475°F (245°C). Place your cast iron pizza pan in the oven to heat for 5 minutes.
Punch down the risen dough and transfer to a floured surface. Roll or stretch the dough into a circle about 12 inches in diameter.
Carefully remove the hot cast iron pan from the oven and brush the bottom and sides with olive oil.
Place the dough into the pan, pressing it to fit the bottom and sides.
Spread the pizza sauce evenly over the dough, leaving a small border around the edges.
Sprinkle the shredded mozzarella cheese over the sauce.
Distribute the sliced mushrooms, green bell pepper, and black olives evenly on top of the cheese.
Sprinkle dried oregano over the toppings.
Place the pan back into the oven and bake for 12-15 minutes, or until the crust is golden and the cheese is bubbly and slightly browned.
Remove the pizza from the oven and let it cool for 3-5 minutes before slicing and serving.