Ok so what I’m about to say does come with the caveat that I do the pricking a hole in the bottom before cooking and also pouring cold water on immediately after boiling thing so maybe that’s why it works for me but honestly I think this will work even if you don’t. Your problem is that you want to remove the membrane between the shell and the egg, WITH the shell, otherwise it becomes very hard to peel in one go and it takes little chunks of cooked egg white with it at variable depths.
To do that, I smack the top and bottom of the boiled egg on a hard surface to precrack the shell at those two places, then I roll the whole egg very gently on it’s side on that hard surface to create little cracks allover, it’s important not to press too hard especially if it’s softer boiled because it’ll just bisect equatorially with the shell still stuck to both halves in broken fragments. Then, the crucial bit, once you’ve precracked the egg all over, you HAVE to start peeling at the bottom of the egg, that’s the fatter end. There should be an air pocket in there that the precracked shell has sorta of collapsed in to, but hasn’t broken off in shards because it’s all held together by this membrane. If you pinch that loosened cracked eggshell at the bottom between your thumb and index finger to gather and collect it, you can kind of pull it up and off to the side a bit which will cleanly make a tear in the membrane allowing you to just kinda push the rest of the shell off in one piece. Because you can just sorta ease it off, sometimes it likes to come off in a nice big chunk like a jacket, sometimes you need to do a continuous spiral but as long you did that pinch technique you’ll be pulling the membrane off at the same time as the shell attached to it and in so doing you don’t have to pull off little bits of shell by themselves it’s one continuous piece and it can’t take chunks of the actual cooked egg with it like happened in your picture.
Ok so what I’m about to say does come with the caveat that I do the pricking a hole in the bottom before cooking and also pouring cold water on immediately after boiling thing so maybe that’s why it works for me but honestly I think this will work even if you don’t. Your problem is that you want to remove the membrane between the shell and the egg, WITH the shell, otherwise it becomes very hard to peel in one go and it takes little chunks of cooked egg white with it at variable depths.
To do that, I smack the top and bottom of the boiled egg on a hard surface to precrack the shell at those two places, then I roll the whole egg very gently on it’s side on that hard surface to create little cracks allover, it’s important not to press too hard especially if it’s softer boiled because it’ll just bisect equatorially with the shell still stuck to both halves in broken fragments. Then, the crucial bit, once you’ve precracked the egg all over, you HAVE to start peeling at the bottom of the egg, that’s the fatter end. There should be an air pocket in there that the precracked shell has sorta of collapsed in to, but hasn’t broken off in shards because it’s all held together by this membrane. If you pinch that loosened cracked eggshell at the bottom between your thumb and index finger to gather and collect it, you can kind of pull it up and off to the side a bit which will cleanly make a tear in the membrane allowing you to just kinda push the rest of the shell off in one piece. Because you can just sorta ease it off, sometimes it likes to come off in a nice big chunk like a jacket, sometimes you need to do a continuous spiral but as long you did that pinch technique you’ll be pulling the membrane off at the same time as the shell attached to it and in so doing you don’t have to pull off little bits of shell by themselves it’s one continuous piece and it can’t take chunks of the actual cooked egg with it like happened in your picture.