Whisk eggs, milk, Buffalo sauce, and salt together in a shallow bowl. Soak bread in the egg mixture until completely absorbed, at least 30 minutes.
Melt butter in a skillet over medium heat until it starts to foam. Arrange soaked bread in the melted butter; sprinkle Parmesan cheese over bread, patting it into the holes of the bread. Fry until bottoms are browned, 3 to 4 minutes. Flip and sprinkle more Parmesan cheese on second side. Cook until bottoms are browned, 3 to 4 minutes. Flip again and cook until cheese is crisp, 2 to 3 minutes.