Sprinkle a small amount of water on a large sheet of plastic wrap. Place 2 of the steaks on top of the plastic and sprinkle again with water. Cover with another sheet of plastic wrap and pound with a rolling pin or meat pounder until about 1/4 inch thick. Repeat with the remaining 2 steaks.

Mix the flour with the salt and pepper in a shallow pie plate. Heat half the oil in a large skillet over medium-high heat until almost smoking. Working in 2 batches, place half the pork in the flour mixture and turn to coat on all sides. Shake off the excess flour and add to the skillet. Cook until lightly browned and cooked through, 3 to 4 minutes per side. Transfer to a plate or platter and cover loosely with foil. Repeat with the remaining oil and pork.

Add the butter to the skillet and heat until melted and foamy. Add the mushrooms and cook, stirring often, until almost all the liquid that the mushrooms give off has evaporated, about 5 minutes. Pour in the Marsala, increase the heat to high, and cook, stirring, until almost all the liquid has evaporated. Add the chicken stock and simmer for 3 minutes. Return the pork to the skillet and simmer, turning often, until the pork is warmed through and the sauce is thickened, about 2 minutes. Transfer the scaloppini to warm serving plates. Stir the sour cream and chives into the sauce. Season with salt and pepper and spoon over the pork.