Place 3 cubes of bread and three cubes of cheese on as many skewers as necessary to use up all bread and cheese. Set aside.
In a large, heavy-bottomed skillet, heat 1 cup oil until almost smoking. Dip each skewer in the egg to coat and cook in the oil until light golden brown. Set skewers of a plate lined with paper towels to drain.
In a small saucepan, heat the remaining 1/2 cup of oil with the anchovies until the oil is boiling and the fish pieces have fallen apart, about 10 minutes. Add the parsley to the oil, pour the oil over the spiedini and serve.