。
She asked us if we wanted to cook.
Literal
She [topic-は] us [to-に] cooking [object-を] want-to-do [whether-か] [quotative-と] asked.
An embedded yes-no question marked by か, then handed off via the quotative と to the verb of asking (たずねる). Structure: '[would you like to cook?]' said-as-a-question + '[said-and-asked].' The ~たい on 料理する is the desiderative — but third-person desire is typically expressed with ~たがる ('appears to want'). Here ~たい sits inside an embedded question, so the speaker isn't asserting her wants directly — they're being inquired about. たずねる is the more formal counterpart of 聞く for 'ask.'