Q: When I go to peel hard-boiled eggs,  I end up with very small, messy-looking  eggs, with big chunks of the white stuck to the shells. How do I get round this?

Older eggs are easier to peel than new ones.  As eggs age, some of the carbon dioxide contained in the albumen escapes, reducing its acidity.  Research shows that the reduced acidity helps with peeling¹.  The trade-off, however, is that in older eggs the yolk tends to move further from being centered.  This happens because the white gets thinner and is less able to hold the yolk in place.  The best compromise is to use eggs that have been stored on their sides in the refrigerator for 7 to 10 days.  Contrary to what you would expect, keeping them on their side results in a more centered yolk than if they are stored on end.

Boil eggs at just below the simmer for about 12 minutes (see "Boiling Eggs without Cracking the Shells"), then plunge them immediately into cold water.  Once they are cooled, peel the eggs.  I find it works better if they are peeled immediately, rather than being stored in the shell.

To peel them, you first need to crack the shell all over.  An easy way to do this is to put the eggs back into the empty saucepan and rattle them around until the shells are cracked.  Alternately, you can cup the egg gently in you hand and crack the shell by tapping it gently against the inside wall of your "impeccably clean" sink.  As you rap the eggs against the sink, let it roll in you hand so that various spots get hit.  In either case, don't strike the egg too hard.  You want to crack the shell while trying not to damage the white underneath.  Rolling the egg around between your hands helps to get the shell cracked all over.

Then peel the shell away.  There is a thin skin under the shell that you want to have come off, too.  Once you start to peel away the skin, it will take the cracked shell with it.  Again, I find it easier to work under gently flowing water, as it washes away loose pieces of shell and helps to lift the skin as the water runs in under it.

Peeled eggs can be stored for a few days in the refrigerator.  Unpeeled hard-boiled eggs will keep for about a week.

¹  Cookwise: The Hows and Whys of Successful Cooking by Shirley O. Corriher.