Dissolve sugar and yeast in warm water in a large bowl; let stand for 5 minutes. Stir in olive oil.
Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine 25 ounces (about 2 1/2 cups) flour, cornmeal, and salt in a bowl. Stir flour mixture into yeast mixture until dough forms a ball. Turn dough out onto a lightly floured surface. Knead until smooth and elastic (about 5 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel sticky).
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 45 minutes or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.) Punch dough down; cover and let rest 5 minutes. Roll dough into an 11 x 15–inch rectangle on a lightly floured surface. Place dough in a 13 x 9–inch baking dish coated with cooking spray; press dough up sides of dish. Spread 1 1/2 cups cheese evenly over dough. Arrange chopped sausage evenly over cheese.
Preheat oven to 400°.
Chop tomatoes; place in a sieve. Stir in oregano and basil; drain tomato mixture 10 minutes.
Heat a large nonstick skillet over medium heat. Coat pan with cooking spray. Add mushrooms to pan; cook 5 minutes, stirring occasionally. Stir in bell peppers; cook for 8 minutes or until tender, stirring occasionally. Arrange vegetables over sausage; spoon tomato mixture evenly over vegetables and sausage. Sprinkle evenly with remaining 1/2 cup cheese. Bake at 400° for 25 minutes or until crust browns and cheese bubbles. Cool 5 minutes before cutting.
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