You will want to use the best quality eggs you can possibly find for this -- the reason is the color of the yolks. Most factory farmed eggs have pale yolks that tend to be watery. You want that golden hue. Size also matters, too, which is why I use duck and goose eggs instead of chicken eggs. Larger egg yolks are easier to hold and they last longer as you grate them over time. But there's no reason not to use a chicken egg if that's all you have. My suggestion for the leftover egg whites is to use them to make pasta, which is what I did in the picture above.
Get yourself two containers: One for the whites, so you can do something with them later, and one to cure the yolks. You will need to lay down a half-inch layer of kosher salt in the bottom of your curing container; you can go a little deeper if you want. Make little depressions in the salt to hold the egg yolks.
Crack the eggs and separate them. Gently lay the yolk in one of the depressions and repeat until you have all your eggs in the container. Now bury them in more salt.
Keep the yolks buried in the salt for a week in the refrigerator. Take them out -- the yolks will be firm and a little tacky still -- and carefully brush off the salt. You might need to remove the salt with a damp paper towel.
Wrap the yolks loosely in cheesecloth and hang them in the fridge until they are dry, about 7 to 14 days. Store in the cheesecloth in a closed container in the fridge.
Notes
Prep time does not include curing time. Once made, these cured egg yolks will keep indefinitely in the fridge.