Is it obligatory to wash chicken eggs and say "Bismillah" over them before cooking and eating them?

There is no obligation to wash eggs before cooking, because the moisture on them is pure according to the majority of scholars, especially if the chickens' feed is pure. However, washing them is better for the sake of cleanliness. As for reciting Bismillah before cooking, it is not obligatory but rather recommended before undertaking any significant matter, based on the Prophet's (peace be upon him) saying: "Every significant matter that is not begun with the remembrance of Allah and 'Bismillahir-Rahmanir-Rahim' is incomplete." However, some scholars have weakened this hadith.