Pound the veal slices to tenderize them. Cut the tenderized slices into about 2½" x 2½" pieces and set aside.
Put flour, salt, pepper and oregano into a large plastic zip baggie and mix well. Add the veal to the bag and shake well until the veal pieces are well coated with the flour mixture. Set aside.
Heat the butter and olive oil in a deep non-stick skillet and saute the onions lightly. Remove the onions from the skillet with a slotted spoon.
In the same pan, brown the coated veal on both sides, being careful not to over-cook. You may have to brown the veal pieces just a few at a time. When the veal is browned, return it all to the skillet with the sauted onions.
Add the Marsala wine and basil to the pan. Simmer until a rich sauce forms. If the sauce is too thick, you can add some chicken broth.
Add the mushrooms and simmer for another 2 to 3 minutes and serve.