Heat oven to 350° F. Coat a 13 x 9 x 2-inch baking dish with nonstick cooking spray.
Heat oil in a large nonstick skillet over medium-high heat. Add onions, peppers and garlic. Cook 7 minutes, stirring so garlic doesn't burn. Stir in soy crumbles, tomato sauce, vinegar, soy sauce and sugar; cover and simmer on medium-low heat for 10 minutes.
Line bottom of prepared dish with half the flatbreads; tear to fit if needed. Spread half the skillet mixture over breads; sprinkle with 1/2 cup of the cheese. Layer with remaining breads and skillet mixture. Cover with foil.
Bake, covered, at 350° F for 25 minutes. Top with remaining 1/2 cup cheese and bake uncovered 5 more minutes to melt cheese. Cool slightly before serving.