What is the proper manner in which to peel an egg on Shabbat?
Rabbi Chayim Palachi (Turkey, 1788-1868), in his work Kaf Ha'chayim, held that one may not handle an egg peel at all on Shabbat, and that t is therefore forbidden to peel an egg in an ordinary manner. Instead, one should drop the egg on a dish such that the shell cracks, and then somehow remove the egg from the shell.
Most other authorities, including the Ben Ish Chai (Rabbi Yosef Chayim of Baghdad, 1833-1909) and Chacham Ovadia Yosef, disagree, and allow peeling an egg normally on Shabbat. However, a certain problem arises due to the Muktzeh status of the shell. Since the shell serves no functional use after it is removed from the egg, as it is unsuitable for human and even animal consumption, it is deemed Muktzeh and one may not move it on Shabbat. Hence, if one places an eggshell into a dish as he peels an egg, the dish, which serves as a "base" for the Muktzeh item, itself becomes Muktzeh. Now Halacha forbids rendering a functional utensil Muktzeh on Shabbat, a prohibition known as "Mevatel Keli Mei'heichano." (Rashi, in two different contexts in Masechet Shabbat, suggests two different reasons for this prohibition. In one context, he explains that rendering an item Muktzeh resembles "Boneh," building, in that the item may no longer be moved on Shabbat, and it thus becomes "affixed" to its place. In the second context, Rashi explains that this resembles destroying a utensil, as it is now rendered unusable on Shabbat.) Therefore, one may not place an eggshell in an empty utensil on Shabbat. Instead, as Chacham Ovadia Yosef writes (Halichot Olam, vol. 3, p. 211), one should first place a permissible item, such as some edible food, in the plate, and then place the eggshells in the utensil. The utensil thus becomes a "base" for both permissible and Muktzeh items, which Halacha allows moving on Shabbat.
Of course, one must ensure not to move the eggshells after they are removed from the egg, as they are considered Muktzeh on Shabbat.
Although peeling an egg is permissible on Shabbat, it may be done only in close proximity to the meal, within a half-hour or so of the meal. One may not peel an egg on Shabbat morning, for example, in preparation for Se'uda Shelishit; an egg may be peeled only shortly before it will be served.
Summary: One may peel an egg normally on Shabbat, but only shortly before one's meal. The eggshells may not be placed into an empty utensils, and one may not move the eggshells after they are removed from the egg.