Prepare the dough first by sprinkling the yeast over a bowl filled with the warm water and stir to combine. Let sit until frothy, about 15 minutes. In the bowl of a large food processor fitted with a dough blade, combine flour, cornmeal, sugar, and salt. Process in water, yeast, and melted butter into the flour mixture until a smooth dough forms. Coat a large bowl with the olive oil. Add the dough, turning to coat. Cover and let rise until doubled, about 1 hour.
Prepare the tomato sauce by melting 2 tablespoons butter in a saucepan over medium heat. Once melted, stir in onion, oregano, and salt. Once the onion is golden, add the garlic and fry until fragrant, about 30 seconds. Add the tomatoes and sugar. Increase heat to medium high to bring the tomatoes to a simmer, then reduce heat to medium low. Continue to cook, stirring occasionally, until the sauce reduces to about 2 1/2 cups, 25 minutes. Remove the pan from heat before adding basil and olive oil. Season with pepper and store in refrigerator until ready to use.
Once risen, divide dough in half. Roll one half into a 8x6" rectangle. Spread 2 tablespoons of softened butter over the rectangle, leaving 1/2" around the sides. Roll the dough tightly, short end to short end, and place seam side down. Flatten the dough into a 9x2" rectangle. Fold 1/3 (short sidinto the middle, then the other side over the first side to form a square. Pinch together the seams and form into a ball. Repeat with other half of the dough and place both back into bowl. Cover and refrigerate until almost doubled, about 45 minutes.
In a large skillet, heat 1 teaspoon olive oil over medium high heat. Add onion and cook until softened and starting to golden. Add sausage and cook, stirring and breaking up into small pieces, until browned. Reduce heat to medium and stir in peppers and mushrooms. Cook, stirring occasionally, until vegetables soften. Drain any excess moisture and set aside.
Preheat oven to 425 degrees Fahrenheit. Grease 2 (9") round pans, with at least 1 1/2-2 inch sides, with 2 tablespoons olive oil each. Roll each ball of dough into a 13 inch circle, 1/4 inch thick. Place in each pan and work the dough into the corners and up at least 1 inch on all sides.
Cover the bottom of each pizza with Mozzarella. Top with a layer of sausage and vegetables. Cover with tomato sauce and top with Parmesan cheese. Bake in preheated oven until crust is golden brown and pizza is heated through, about 30 minutes. Let settle about 10 minutes before serving.