Pound the chicken breasts thin using a meat mallet and placing them between two pieces of plastic wrap. Pat them dry using paper towels.
Prepare the flour for dredging by mixing it in a shallow dish with the Italian seasoning, dry parsley, salt, and pepper.
Place the chicken breasts one by one into the seasoned flour (cover them on both sides), and remove the excess.
Melt the butter with the olive oil in a large skillet over medium-high heat and brown the chicken breasts for about 2 minutes on each side.
Place the sliced onions and mushrooms over the chicken breasts, pour in the wine, put the lid on, and simmer for 15 minutes. Add salt and pepper to taste.