Heat the oven to 200 degrees F. In a large skillet, heat 1/2 cup olive oil and enough vegetable oil to reach a depth of 1-inch. Meanwhile, prepare the sandwiches: Place 1 slice of bread on work surface. Top with 2 slices mozzarella. Cover with a second slice of bread. Repeat with remaining bread and cheese to make a 4-layer sandwich. Place a 10-inch skewer through the center of each sandwich quarter. With a serrated knife, gently remove the crusts and discard. Cut sandwich between skewers into 4 equal quarters.
Combine eggs, milk, salt, and pepper. Whisk, and pour into a shallow baking dish. Dredge sandwich quarters in flour and shake off excess. Dip in egg mixture turning to coat on all sides. Transfer immediately to hot oil. Fry, turning occasionally, until golden on all sides. Drain on a paper towel-lined baking sheet and place in oven to keep warm.
In a medium skillet, heat remaining 2 teaspoons olive oil over medium-high heat. Add garlic and cook until golden. Add anchovies and cook stirring until they almost disintegrate. Stir in white wine, lemon juice, and capers. Bring to a vigorous boil and continue to boil until reduced by 1/2.
Stir parsley into reduced sauce. Remove skewers from sandwiches and serve with sauce spooned lightly over sandwiches. Garnish with parsley sprigs.
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