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.
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.