L’ALBI, Spain—Just off Central Park, chefs at one of New York’s hottest Italian restaurants are pondering how to handle surging prices for a vital ingredient: olive oil.

Marea and other restaurants specializing in Mediterranean food are feeling the pinch as a drought in Spain, by far the world’s biggest producer, pushes up global prices. And with Americans pouring increasing quantities of olive oil onto their salads and into their frying pans in recent years, many households are likely to notice rising costs over time, too.