Preheat oven to 350 degrees.
Cream butter well. Gradually add to it the oil and sugar. Continue beating until well creamed. Add to this the orange juice and brandy and mix well.
Sift the dry ingredients (using about 3 cups of the flour). Add to the creamed mixture gradually. Slowly add the rest of the flour. (Add just enough flour so that finiki will hold its shape.)
Form into oval shapes, putting each finiki onto a grater and indenting with fingers. Turn over and place on ungreased cookie sheet.
Bake at 325 - 350 degrees for 25 to 35 minutes. Don't overcook - don't let them get browned.
When finikia are cool, dip into hot syrup (see below) for about 4 minutes,
roll finikia over and let sit another 3 or 4 minutes, remove from syrup
with a slotted spoon, put on a clean plate and then immediately roll in
finely chopped walnuts and put on clean plates.
Bring all ingredients to a boil and simmer for 15 to 30 minutes (skim the brown froth off the top). Remove from heat and cool for 3 or 4 minutes. Put back on burner on WARM or lowest setting and dip the finikia.