Prepare the chicken: If you're using 2 large chicken breasts, cut them in half lengthwise since they may be too thick. Place the chicken breasts in between plastic wrap and pound them with a meat mallet so that they're about 1/2 inch in thickness. Make sure not to break the meat.
Dredge the chicken: Add the cup of flour to a shallow plate, add salt, pepper, oregano and mix well. Dredge each chicken breast through flour and set aside.
Cook the chicken: In a large skillet heat 1 tbsp of olive oil with 1 tbsp butter. Add a couple pieces of chicken at a time, you don't want to crowd them, and cook until it starts to brown, about 3 minutes per side. It's ok if it's not cooked through completely. Repeat with remaining chicken.
Form the base of the sauce: In the same skillet add the other tbsp of olive oil. Add garlic and prosciutto and cook for about a minute. Add mushrooms and cook for 2 minutes or until they start to brown slightly.
Finish the sauce: Add 1 tbsp of flour and stir. Pour wine and chicken broth to the skillet and stir and make sure you scrape the bottom of the pan. Season with salt and pepper if needed. Let it simmer for about 3 minutes. The sauce will start to thicken a bit.
Complete the dish: Add chicken to skillet and cook for another 5 minutes until sauce reduces and thickens a bit more and the chicken cooks through if needed. Garnish with parsley and serve.