Divide half of the cheese equally among 6 bread slices. Top with the basil leaves then the remaining cheese. Cover with the remaining bread slices, pressing gently, to form 6 sandwiches.
Whisk the eggs, salt and pepper in a large bowl until well blended and frothy. Place the flour in a medium bowl. Add enough oil to a heavy large frying pan to come 1/2-inch up the sides of the pan. Heat the oil over medium heat. Dredge the sandwiches in the flour to coat lightly, then dredge them in the egg mixture to coat completely and place in the hot oil. Cook until the cheese melts and the sandwiches are golden, about 3 minutes per side. Cut the sandwiches into triangles. Transfer the sandwiches to a platter and serve with the pesto.