Heat butter and olive oil in a wide skillet over medium heat. Add the sliced onions and cook with the lid on, stirring occasionally, until the onions are deeply caramelized and jammy, about 30-40 minutes.

In a food processor, combine the softened cream cheese, sour cream, mayonnaise, Worcestershire sauce, salt, and black pepper. Pulse until smooth and well combined.

Add the caramelized onions to the food processor and pulse a few more times until just combined. Chill the onion dip for at least 30 minutes to allow it to firm up and the flavors to meld together.

Transfer the dip to a serving bowl and let it sit at room temperature for a few minutes before serving. Garnish with chopped chives and serve with potato chips.