Why an egg was the traditional choice, and the one common mistake that ruins it.
Q. Eruv Tavshilin when Rosh Hashana falls on Thur. and Fri.?
One isn’t allowed to prepare for Shabbat on the holiday, for each one deserves its own significance and shouldn’t be seen as a day of preparation for something more important. Accordingly, when a holiday comes out on Fri., or if Rosh HaShana falls on Thur. and Fri., in order to allow doing so, once must prepare the Eruv Tavshilin before the holiday (in our case of Rosh HaShana, on Wed.), and recite the proper blessing. By expressly preparing (through buying, or even better, cooking on Wed. itself) something cooked and something baked for Shabbat, already before the holiday, the halacha allows that family to “continue” cooking and baking on Fri., for the upcoming Shabbat. The cooked dish must be one that is often eaten with bread, for example meat or eggs (e.g. egg salad). The custom was to use eggs, for they could last the two and a half days (from Wed. to Sat.) without spoiling, even before there was refrigeration. However, one must be careful not to peel the egg before Shabbat, for one isn’t allowed to eat an egg or onion which was left unpeeled overnight, unless at least some of the peel remains, or unless one mixes the peeled egg with other ingredients, like salt or mayonnaise. Don’t forget to eat the Er. T. on Shabbat. Only if one accidentally forgot to make an Er. T., can he rely on the local rabbi who has in mind all those who forgot.