。
How should I cook the eggs?
Literal
Egg [topic-は] how cook-shall-we-[question-か].
~ましょうか ('shall I/we?') is the polite volitional used to offer, invite, or ask for instructions. 調理する ('to cook / prepare food') is more formal than 料理する, typical of professional kitchen contexts or waitstaff addressing customers. どのように ('how / in what way') is the formal question equivalent of the casual どう.