what I usually do is take them straight off of the burner and set the pan right in the sink just like my mom did. and run cold water into the pan until the eggs cool. once cool not warm I hit the small end of the egg on the counter and then roll to make the shell shatter. then just gently pick a spot and start peeling. I usually don’t have any issues at all, as long as they are cool enough.
I used to have to peel dozens of eggs when I worked in a deli. Start with cold water, take a tack and pierce a hole in the large end of the egg. Bring to a boil then cover and remove from the heat. Let them sit for 10 minutes in the pan. Meanwhile, make a bath of ice water and immediately put the eggs in the bath of ice water after they have finished cooking. Let them cool completely then peel. Good Luck!
I cover the eggs with cold water, bring to a light boil over medium heat, immediately turn off heat and cover with lid, don’t touch for 15 minutes. After 15 minutes, remove lid, run cold water over eggs, after they’ve cooled enough to handle, bash large end of egg against sink, then continue to run cold water over them. Always easy to peel, starting with large end.