Place about 1/3 cup of potato starch, or all-purpose flour in a bowl. Pat chicken breasts dry with a paper towel and then dip each piece lightly into the starch. Shake off any excess.
Heat the oil in a large skillet over medium-high heat until very hot but not smoking. Add 1 tablespoon of the butter and cook the chicken breasts until golden brown on both sides, about 3 minutes per side.
Transfer to a plate and set aside.
Add 1 tablespoon of the remaining butter to the pan and add the scallions and mushrooms. Cook, stirring frequently, until mushrooms are golden brown around the edges and have given off their liquid.
Add the Marsala wine and vanilla extract and bring to a boil, scraping to remove any browned bits from the bottom of the pan. When the wine has reduced by half, add the chicken stock and cook for 3 minutes, or until the sauce has reduced slightly.
Lower the heat to medium and return the chicken breasts to the pan and continue to cook until they are cooked through and the sauce has reduced, about 5 to 6 minutes. Swirl in the remaining 2 tablespoons of butter, add salt and pepper, to taste. Garnish with chopped parsley or chives and serve immediately.