Breakdown of Möchtest du Käse oder Schinken?
du
you (informal singular)
der Käse
the cheese
möchten
would like
oder
or
Questions & Answers about Möchtest du Käse oder Schinken?
Why is there no verb like haben or essen at the end of the sentence?
While möchten (would like) often acts as a helping verb paired with an action verb at the end of the sentence, it can also be used all by itself with a noun. Just like in English, you can simply ask "Möchtest du Käse?" (Would you like cheese?) without needing to add haben (to have) or essen (to eat).
Why isn't it "den Käse" or "einen Käse"?
Käse and Schinken are the direct objects here, so they are technically in the accusative case. However, because we are talking about an unspecified amount of food or a general category, German drops the article entirely. This works exactly like English, where we offer "cheese" instead of "a cheese" or "the cheese".
Could I say "Magst du Käse oder Schinken?" instead?
You could, but the meaning changes. "Möchtest du..." means "Would you like...?" and is used to offer someone food right now. "Magst du..." means "Do you like...?" in general, asking about their tastes rather than offering them something to eat.
How would this sentence change if I were speaking to someone formally, like a customer?
The verb ending would change to match the formal "you" (Sie). Instead of the informal "Möchtest du...", you would use the formal version: "Möchten Sie Käse oder Schinken?"