This was an experiment to re-create a dish from my youth.

12 eggs
3 large beets
1 large onion
1 cup sugar
2 cups vinegar
2 cups water
Special equipment: a wide-mouth 1-gallon glass jar.

We normally get our eggs from the local farmer down the road, however he was 
out so we picked up a carton from the supermarket.  Next time, I'll use farm 
eggs!  The beets and the onion were from our Farm-to-Table program and local 
grown.  The vinegar was a home-made apple cider vinegar.  The water was from 
our well, which means next time we do this, only the sugar will be 
store-bought.

Boil the eggs, rinse them, let them cool.  While the eggs are boiling, peel 
and thinly slice the beets and the onion.  I used the onions and beets raw.  
Various recipes on the net for pickling onions say boil them for three minutes 
with skins on.  I may try boiling the onions and beets next time.  When the 
eggs are cool, shell them but leave them whole.  In the clean glass gallon jar, 
put a few eggs, a layer of onion and beet slices, a few eggs, a layer of onions 
and beets and keep alternating until either the jar is full or you're out of 
eggs, onions and beets.  I lucked out and had just enough to fill the jar.  
Boil the water, sugar and vinegar on the stove.  When it reaches full boil, 
shut it off and let cool for a minute.  Pour the still hot liquid into the 
glass jar over the eggs, onions and beets.  Seal the jar.  Store the jar in a 
cool dark place for at least three days.  Do NOT put the glass jar in the 
fridge until it has cooled off, the difference in heat between the inside and 
outside of the jar can shatter the glass. 

After a week, I fished out an egg and bit into it.  It tasted exactly like the 
eggs of my youth and it was purple all the way thru to the yolk.  The onions 
and beets were pickled as well and perfectly edible if a bit crunchy.  Next 
time I'll boil them to see which way I prefer.  I was able to find references 
on the net indicating that medieval peasants frequently pickled all kinds of 
foods including eggs to preserve them for long periods, however I all the 
references to the history of beet pickled eggs kept leading back to 
Pennsylvania Dutch history.  I distinctly remember them as being a traditional 
Polish food.