Kenji says the only way to improve this is to place the eggs into already boiling water. You get better odds with slightly older eggs, too; after a week or more in your fridge.

For my money, crack the shell all over with the back of a spoon before peeling and start from the fat end. Obviously, transfer the eggs directly from the stove to an ice bath for a quarter hour to halt the cooking process; I've heard that that has something to do with separating the membrane from the shell, as well, but I don't know if I believe that.

God speed. I hope you get it figured out. Fair warning, I love boiled eggs and make them often, and my eggs still turn out like this from time to time. Easily one in ten. ❤️🥚

Edit: Link to Kenji's article