Halve chicken breasts and pound into 1/4-inch thick cutlets. Add flour to a large shallow dish. Dredge chicken cutlets in flour, shaking off excess.
In a large skillet set to medium-high heat, melt butter and olive oil. Once hot, add chicken cutlets and cook until golden brown, about 5 minutes per side. (Chicken does not need to be completely cooked, as you will continue cooking later in the recipe.) Remove cutlets from pan; set aside.
Reduce heat to medium. Add mushrooms and garlic. Sauté for about 3 minutes, until mushrooms are slightly limp. Add white wine and thyme, scraping up browned bits in bottom of pan. Return chicken to pan and simmer for 15-20 minutes.
Remove chicken from pan (Yes, one more time!). Stir in Parmesan, Dijon and cream until well blended. Return chicken to pan. Serve cutlets with creamy mushroom sauce spooned over top.
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