Heat the oil in a large, heavy skillet over medium-high heat. Add the mushrooms, salt, and pepper, and cook, stirring, until the mushrooms are wilted and begin to caramelize. Add the shallots and garlic, and cook, stirring, for 30 seconds. Add the wine and cook, stirring to loosen any browned bits in the pan, and until the liquid has almost all evaporated, about 2 minutes. Remove from the heat and transfer the mushrooms to the bowl of a food processor. Add the cheese, bread crumbs, parsley, basil, and oregano, and process on high speed to a thick paste.
Transfer to a bowl and divide into 4 equal portions. Roll each into firm balls and use to stuff the chicken breasts.
Yield: about 1 cup