In a mixing bowl, whisk eggs until well beaten, then whisk in milk, vanilla, and melted butter.
Whisk in flour and sugar. Whisk until smooth, and almost no more lumps show - that's ok if you still have some small lumps in the batter, but avoid big lumps for a tasty thin crepe.
When the pan is hot, lightly oil it with coconut oil or avocado oil. You can use an oil spray or rub a piece of oiled absorbent paper on the pan. Don't overoil the pan, or you will fry the crepes.
Pour about 1/3 cup of batter in the pan, tilting the pan into circular motion - see my pictures above in this post for visual help. This slowly covers the pan surface with the batter. If the pan is hot enough, the batter should stick well to the coating. If it splits in the pan, it means the pan is too hot.
Cook for 2-3 minutes on one side or until the sides dry up and you are able to slide a spatula under the crepe to flip it over.
Flip the French crepe and cook for an extra minute on the other side.
Store on a plate while you are cooking the remaining batter.
Serve with chocolate spread, jam, or powdered sugar.