"Tu écris le menu ?"
Yes, inverting the verb and subject ("Écris-tu...") is the formal way to ask a question in French.
Placing the subject before the verb ("Tu écris...") is more colloquial.
"le menu" refers to a fixed price menu at a restaurant, not the piece of paper you read (or write) with the list of food available to order; the correct word should be "carte"
In French, "menu" means:
the list of dishes that are served for a meal (e.g. "menu gastronomique" = "gourmet menu").
the fixed price menu at a restaurant (e.g. "menu enfant" = "children's menu")
the piece of paper/cardboard with h list of dishes (in English "menu (card)").