。
How should I cook the eggs?
Literal
Egg [topic-は] how cook-shall-we-[question-か].
どう ('how') is the casual question word, less formal than どのように. 料理する is the everyday verb for 'cook,' contrasting with the more formal 調理する used in professional kitchen contexts. The polite volitional ~ましょうか offers to take action — 'shall I cook them how?' — neither demanding nor presumptuous.