Heat the olive oil and 2 tablespoons of the butter in a large, heavy skillet over medium heat until the butter is foaming. Add as many of the scallopine as will fit without touching and cook until golden brown on the underside, about 3 minutes. Flip and cook until the second side is lightly browned, about 2 minutes. Remove from the pan and repeat with the remaining scallopine, adding more oil to the pan if necessary. Remove all scallopine from the pan when browned and drain on paper towels.

Drain the fat from the skillet and return the pan to the heat. Add 2 tablespoons of the remaining butter and, when it is melted, stir in the shallots. Cook, stirring, until wilted, about 3 minutes. Scatter the mushrooms into the pan, season them lightly with salt and pepper, and cook until the mushrooms are lightly browned, about 4 minutes. (If the mushrooms begin to give off liquid, you'll have to wait for that to evaporate before the mushrooms begin to brown.)

Pour in the Marsala, bring it to a boil, and cook until the Marsala is slightly syrupy, about 3 minutes.

Add the remaining 4 tablespoons of butter and pour in 1 cup chicken stock. Bring to a vigorous boil and season lightly with salt and pepper. Boil until reduced by half, then tuck the scallopine into the sauce. Cook, turning the scallopine in the sauce, until the scallopine are heated through and the sauce is thickened enough to lightly coat them, about 3 minutes. If necessary, add small amounts of chicken stock to make enough sauce to generously coat the scallopine. Swirl in the parsley and serve on hot plates, dividing the scallopine among the plates and spooning some of the sauce and mushrooms over each serving.