All of the following comments and directions come from How to Poach Eggs for a Party | The Food Lab
"Here's a secret: When poaching eggs, you don't have to cook them to-order. In fact, you can poach them up to five days in advance with no loss in quality. Not only that, but it takes just 2 minutes and zero skill to take those eggs from fridge-cold to liquid-yolk-ready-to-serve once brunch begins."

Eggs
Water

White vinegar
😂

Poach as you usually do. Or look at this: Instant Pot Poached Eggs. Or this: Sous Vide Poached Eggs.

After three to four minutes, the egg white should be fully set with the yolks still tender.

You can test this by gently picking one up with a slotted spoon and poking the yolk with your fingertip.

It should feel like a plump water balloon.

Immediately transfer the eggs to a bowl filled with very cold water (you can add ice if your tap water is not cold enough).

Repeat the cooking step as many times as necessary until you have all the eggs you need.

Once all of your eggs are chilled, you can transfer them to a sealable container and top them up with cold water.

Cover and store in the fridge for up to five days.

 

1) Water should have tiny bubbles at the bottom, NOT a rolling boil.

2) Put a tiny bit of white vinegar in the water. Not too much!

3) Eggs shouldn't be fridge cold. Heat them up in hot tap water. This will help them to set and not have so much egg white going all over the place.

Ready to serve those eggs? Good.

Here's the fun part: you don't even need to boil water.

In fact, it's better not to.

With a pot of simmering water you run the risk of accidentally overcooking them.

The water that comes out of your hot tap will almost certainly be hot enough to re-warm the eggs without overcooking them.

130°F - 140°F is just right.

Let them sit for 2 minutes to warm up, topping up with more hot water as necessary (this is mainly an issue if you are reheating lots and lots of eggs).

When ready to serve, drain off the excess water by transferring the eggs to a clean paper towel to blot.
ED: I am not sure how "food safe" this is, but I would suggest leaving the eggs on a paper towel BEFORE heating them to get them to room temperature. If you take them right out of cold water and into the hot water, the hot water cools down too much. 
Or make sure your water is a little hotter than 140°.