In a 12-inch oven-going skillet, cook chicken, onion, salt and pepper in 2 tablespoon hot olive oil over medium heat for 10 minutes or until onion is tender and chicken is no longer pink, stirring frequently; drain and return to skillet. Stir in the brown sugar, vinegar and water. Cook and stir over medium heat for 5 minutes more. Remove meat mixture from skillet; set aside.
In the same skillet, heat the unsalted butter and 1 tablespoon olive oil over medium heat until butter is melted, tilting skillet to coat bottom with melted butter mixture. Remove from heat. Arrange red and yellow tomato slices in the bottom of the skillet, overlapping slices as needed.
In a food processor, combine bread crumbs, basil leaves, dried tomatoes and garlic. Cover and process until basil is chopped and mixture is combined. Layer bread crumb mixture over tomatoes in skillet, pressing down lightly. Spoon cooked chicken mixture over bread crumb mixture. Sprinkle with shredded cheese.
Unroll pie crust and, if necessary, roll to diameter large enough to cover mixture in skillet. Lay pie crust on top of layers in skillet.
Bake in a 425° oven for 30 to 35 minutes or until golden brown. Remove skillet from oven. Let stand 5 minutes on a wire rack. Loosen edges; invert skillet over a large serving plate. Garnish with parsley and crumbled cheese. Serve warm.