Lessons Learned...
I tried them again this morning, and taking all of your tips into consideration, tried again. I decided I was boiling the water too much, and using eggs that were about a week old didn't help either. I noticed even when I cracked the newer ones open, the uncooked white part stayed in a little mound better in the bowl!
I had always made them in a microwave-poacher, a little plastic thing that had 2 indentations in them that you cooked as joanie and others suggested. But I LOVE A GOOD food challenge (and found that usually the yolks were too hard, and the whites too rubbery), so I have been trying the conventional way lately.
Well, over the last month and a dozen or so messy poached eggs later, I found:
Put them in a little bowl first, and slide them into the simmering water (one at a time). Then when they roll out, they have a headstart on the shape you want them to have when cooked. When the egg touches the hot water, it sort of immediately gets a white film on it that contains those little "would-be" threads. I needed more water in the pan - 3 inches gives them room to boil and turn as they cook (ah haaaa!). The slow simmer seems to keep them turning slightly. And simmer, don't boil (boiling = foam).
I used to use the "vortex" method, but that's when my little egg white strings turned to a scattering of egg whites. White threads everywhere! And constantly pushing the cooking egg white over the yolk, while it sounded good, just didn't work for me! ;o)
Thanks for the tips!