Preheat the oven to 325.
Place a deep skillet or saucepan over medium-high heat. Add in a drizzle of olive oil and the chicken pieces, skin side down in the oil. Sear for 4-6 minutes until the chicken is lightly golden. Spread the chicken pieces in to a casserole dish and place in the oven for 25-35 minutes until the chicken is firm, but not hard to the touch, the internal temperature is about 165-175 degrees OR the juices run clear when you pierce it with a knife.
Meanwhile, make the sauce:
Using the residual chicken fat, toss in the onion strips. Saute lightly for 8-12 minutes until lightly golden. Add in a splash of the marsala wine. When the onions absorb all of the wine, place it in a bowl to set aside.
Next, add a splash of oil and saute the mushrooms for 5-10 minutes until lightly golden. Again add a splash of marsala, once absorbed add to the sauteed onions.
Next, reduce the heat to medium and add in the remaining olive oil and a tablespoon of butter (or more olive oil instead). Sprinkle in the flour and whisk rapidly to create a roux or paste.
Slowly whisk in the chicken broth. Continue to whisk until it thickens into a nice sauce. Add the remaining marsala. Season with salt and pepper, to taste. Continue to simmer until the sauce is thickened.
Fold in the mushrooms & onions. Serve the chicken with the sauce on top.