In a medium bowl, beat the egg. Whisk in the milk, butter, and vanilla extract if using.
In another bowl, sift together the flour, sugar, baking powder, baking soda, and salt.
Use a wooden spoon or spatula to gently fold the flour mixture into the liquid ingredients until just incorporated.
Cover and refrigerate the bowl for 1 hour.
Place a taiyaki pan on the stove over medium low heat. Lightly grease with vegetable oil.
Once heated, fill one side of the taiyaki pan cavities 1/3rd full with the rested batter.
Add a heaping tablespoon of chocolate hazelnut spread in the center of the fish and cover with batter to fill out the fish shape. Be careful not to overfill. Close the pan and flip.
Cook for about 1-2 minutes until golden. Flip and cook the other side until golden, another 2 minutes, checking the color occasionally once it has had a chance to set.
Transfer the golden taiyaki to a wire rack and repeat with the remaining batter and chocolate hazelnut spread, greasing the pan as needed.