For hard ‘boiled’ I’ve had best luck steaming them. You need one of those put-on-top-of-a-pot steamers (looks a bit like a double boiler but with holes in the bottom)

• put eggs in the pot-top steamer
• bring the water to boil
• put pot-top steamer on pot of boiling water, moving lid from pot to steamer if necessary
• wait around 14 minutes
• dump egg(s) into a bowl of ice water

One nice thing about steaming eggs is that the timing is not dependent on the number of eggs being cooked. I’ve done as many as 8, using the same timing as for one.

Another nice thing is that you are effectively hot-starting the eggs, so they’ll be surprisingly easy to peel (even the ones your hens laid today).

Adjust timing to your own tastes. I’ve found that 14:40 makes a perfect-for-egg-salad egg; yolk a bit crumbly but no sulfur smell or green ring. between 13 and 14 gets you hard cooked with a creamy yolk. Try it with a single egg to adjust timing to your own preference — you can use that timing with multiple eggs.

As far as poaching goes, I once asked a restaurant chef who had just made about the best eggs benedict I’d ever had. He said he uses “no older than laid yesterday” and otherwise does nothing special.

Agreed 100% on French omelettes and scrambled — so long as you have a good pan.