To prepare dough, dissolve yeast and sugar in 1 cup warm water in a large bowl; let stand 5 minutes. Lightly spoon flours into dry measuring cups; level with a knife. Add 2 cups all-purpose flour, whole wheat flour, 1/4 cup cornmeal, 1 teaspoon salt, and 1 teaspoon oil to yeast mixture; stir until a soft dough forms. Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands. 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, 1 hour or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.)
To prepare topping, heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add mushrooms, onion, and garlic; sauté 8 minutes or until mushrooms are tender and liquid almost evaporates. Remove from heat; stir in thyme and 1/2 teaspoon salt.
Preheat oven to 475°.
Punch dough down; cover and let rest 5 minutes. Roll dough into a 13-inch round on a lightly floured surface. Lightly coat a large baking sheet with cooking spray, and sprinkle with 1 tablespoon cornmeal. Place dough on baking sheet. Spread mushroom mixture over dough, leaving a 1-inch border; sprinkle with cheese. Bake at 475° on bottom rack of oven 13 minutes or until crust is lightly brown and cheese melts. Sprinkle with parsley. Cut into 12 wedges. Serve warm or cold.