Can I cook a fairy?

Technically, no, you can't cook them, but you can use them in recipes to get a better result – the fairy will be released after doing this.

From this guide:

And if you throw in a Fairy, you’ll get a lot more hearts from whatever you make together. Don’t worry, it won’t actually cook the Fairy.

This guide confirms this behavior, as well.

Cooking with a Fairy will add considerably to almost any meal you want to make. Don't worry, you won't actually cook the Fairy, but she will hover around the pot and enhance the meal for you. For instance, Fried Egg and Rice made with Hyliran Rice and a Bird Egg usually restores four hearts. However, add a Fairy to that recipe and it restores 14 hearts.

A user here says that using a Fairy in a recipe will result in them flying around the pot and strengthening the result.

You don't actually cook them. They fly around the pot and strengthen the result.

You can get a unique recipe for only cooking a Fairy. It will produce a potion that restores 8 Hearts per Fairy. It also has a unique sprite. However, it does not allow you recover if you lose all your hearts.