Add the remaining 1 tablespoon oil to the pan. Add shallots and mushrooms; cook, stirring frequently until the mushrooms are browned, 2 to 4 minutes. Add wine and the remaining 1/4 teaspoon salt; cook, stirring up any browned bits, until the liquid is mostly evaporated, 1 to 3 minutes. Stir in half-and-half and herbs; cook until bubbling, about 1 minute more. Serve the chops with the mushroom sauce.