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On sheet of wax paper or a paper plate combine flour, Parmesan cheese, and 1/4 tsp each salt and pepper.
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Dredge chicken in flour mixture, coating both sides and reserving any remaining flour mixture.
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In 10-inch skillet heat 1 1/2 tsps oil over medium-high heat; add onions and garlic and saute until onions are softened.
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Add mushrooms, basil and remining 1/4 tsp salt and saute until mushrooms are tender, about 5 minutes.
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Transfer mixture to plate and set aside.
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In same skillet heat remining 1 1/2 tsp oil; add chicken and cook, turning once, until lightly browned 1 to 2 minutes on each side.
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Stir in any reserved flour mixture; gradually add wine and water and, continuing to stir, bring mixture to a boil.
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Return mushroom mixture to pan and cook until heated through.