Preheat oven to 200° F or lowest setting. Using eight slices of bread for each, make two layered sandwiches alternating bread and mozzarella. Cut and fit the mozzarella slices as necessary to cover the bread more or less evenly. The cheese shouldn’t overhang the bread slices, as the crusts will be trimmed off before cooking. Imagine each sandwich cut into four squares, and place a sturdy wooden skewer through the center of each imaginary square. With a serrated knife, cut off the crust on all four sides, then cut the sandwich into four squares, with a skewer at the center of each square. Press gently as you cut, to make nice, compact sandwiches. Repeat with the other sandwich. You now have eight multilayered sandwiches, each on a skewer.

Pour enough vegetable oil into a wide, deep skillet–cast iron is perfect–to fill it by 1½ inches. Heat over medium heat until the oil registers 350°F on a deep-frying thermometer. Line a baking sheet with a double thickness of paper towels and set aside. Meanwhile, in a shallow wide bowl, whisk the eggs, milk, salt, and pepper until thoroughly blended.

When the oil comes to temperature, dredge each skewered square in the flour, being sure to coat all sides lightly but well. Working with half the floured skewers at a time, dip them in the egg mixture to coat on all sides. Don’t soak the bread in the egg, just coat it thoroughly. Let some of the excess egg drip back into the bowl, and gently slip the squares into the oil. Fry, turning as necessary with a pair of long tongs, until golden brown on all sides, about 4 minutes. Drain the sandwiches on the paper-towel-lined pan and place in the warm oven. Repeat with the remaining sandwiches.

Make the sauce: Heat the olive oil and garlic in a small skillet over medium heat. Cook, shaking the pan occasionally, until the garlic is golden. Stir in the anchovies and cook until they dissolve. Stir in the capers and cook till they sizzle. Pour in the lemon juice, then the white wine. Bring the ingredients to a vigorous boil, add the vegetable stock, and boil until the sauce is lightly thickened, about 3 minutes. Stir in the parsley and remove from the heat.

Arrange the hot mozzarella sandwiches on a warm platter or plates and pull the skewers from them. Spoon the sauce over and around the sandwiches and serve immediately.