Place one slice of bread on working surface, then place 2 slices of prosciutto and 2 slices of mozzarella on top of bread and then place another slice of bread on top.
Trim the crusts, cut in half forming 2 rectangles and place 2 toothpicks through each of the rectangular sandwiches to hold together.
Repeat using other 2 slices of bread, mozzarella, and prosciutto.
Dredge each spiedino in flour then dip in egg, covering all sides.
Heat remaining 1/4 cup of olive oil in large sauté pan over medium heat. Heat about 3 inches of vegetable oil in a large sauté pan over high heat.
Place spiedini in pan and fry on both sides until brown.
Place them in serving plates and remove the toothpicks.
Melt 2 tablespoons of butter in a sauté pan over medium heat.
Add anchovy filets and sauté until anchovies are heated through.