how make perfect steamed egg?

Boil them instead. If the water is at a consistent boil, then it can’t go over 212F max otherwise it begins rapidly evaporating. Steam can go well over that temp threshold. I’m not sure if that’s the cause of your uneven eggs, but I imagine it’s a factor. It just gives you a bit more consistency.  Boil about 6m, then cool them in an ice bath, or at the very least running cold water

If you can hear the chick chirping you are cooking the wrong eggs.

I’ve never steamed an egg - didn’t even know that was a thing?? But honestly it sounds like you are on your way to perfection, just keep trying.

If you are really trying to dial it in, use the same burner on the stove, the same setting, the same pot, the same amount of water, everything exactly the same every time and then vary your time. When I’m dialing something in it’s usually a baking recipe and I do it over and over, sometimes every day or every other day, writing every single thing down as I go along so I can intentionally review the recipe while I sample it and make notes for the next tweak.

I’m currently tweaking my beefsteak technique. The amazing New Zealand steaks I used to get from the freezer section at Trader Joe’s got discontinued a while ago, and I’ve just found a new TJs steak that I really like - but I have to figure out how to do it perfect because it’s a little different than the last one.

You have to watch the clock like a hawk or else have a good egg timer and run to get that egg when you hear the bell.

Supposedly the best way is to cover the eggs with cold water, bring the pan to a boil, then take it off the stove.  After 10 minutes of cooking off the heat, but still in the boiling water, plunge the eggs again into cold water to stop the cooking process.  This should also reduce the chance that you will get that weird green ring around the yolks.