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In a small bowl, whisk eggs, seasoning with salt and pepper, to taste. On another plate, place flour, and season with salt and pepper, to taste.
Heat a large nonstick skillet or a griddle over moderate heat. When it is hot, lightly oil the griddle if it is not nonstick. Butter 1 side of each bread slice, then put 4 slices, buttered side down, on a work surface.
Spread 1 side of 4 bread slices with the anchovy mixture, dividing it evenly. Top with the sliced rounds of mozzarella, dividing it evenly. Top each with a second slice of bread, buttered side up.
One at a time, dip the sandwiches in seasoned flour, coating well, then the egg bath. Set them in the skillet or on the griddle. Cook, turning once, until the bread is well browned on both sides and the cheese is molten, about 3 minutes per side. Cut in halves or quarters and serve immediately.