I have never used a recipe , I just used  clear white ( distiller vinegar and hard boiled eggs. You need to use very fresh eggs and hardboiled them well .. But not so as you get that grey ring around the edge of the egg.
You need to leave the eggs a good 3 or 4 weeks before you eat them . Maybe more .
It's been a long while since I made pickled eggs . But I am sure I did not add anything else .
I just seem to remember hard boiling eggs , peeling them
And packing them into a jar then coveting them with distiller vinegar, putting the lid in the jar then thejar id eggs into a cupboard .
I always used a large screw topped jar.

The hard thing is testing the eggs every so often .. You get an egg out of the jar and eat it to see if its picked enough telling yourself that's what needs to be done.
Then you discover that the eggs could do with more time .. So you leave them a bit longer and then feel the need to repeat the testing ...
And so it goes .
Finally the pickled eggs are perfect and just as you like but as you retern the jar to the shelf you look at it and realise that you had just eaten the last  one !
Or you leave the jar alone for several weeks  as I did when one of my daughters was smaller only to get it out if the cupboard to find it half empty as the small daughter had been busy testing the eggs herself !

Pickled eggs are a strange thing really , one of those things that I think of as peculiarly British.
Often can be found in fish and chip shops .
They don't seem to  be as  popular these days as they did 30 years ago.

I like hot brine  over the eggs.  A room temp brine seems to affect the texture of the eggs  making them rubbery , the same goes for using just straight up vinegar instead of cutting it with a bit of water.

amounts depend on the size of container and amount of eggs  but generally  I do 2 parts white vinegar to 1 part water with a  good pinch of salt  brought up to a boil then poured over the hardboiled peeled eggs.   I will often add pieces of onion or whole cloves of fresh garlic as well  and sometimes mustard seeds and black peppercorns.

Thanks for all your tips.  I will let you know how I get on.  Now, why did I get rid of all my large glass jars?

Mummza, it is 30 years since we lived in England.  I wonder what else has fallen out of food fashion since then?  We used to get round sliced luncheon meat with a slice of hard boiled egg in the middle, that was my favourite.