Melt the butter with the olive oil in a large skillet over medium heat. Add the sliced onions, kosher salt, and black pepper, then cook slowly for 30–40 minutes, stirring every few minutes, until the onions are deeply golden, soft, and sweet.

Heat the oven to 375°F. In a mixing bowl, combine the softened cream cheese, sour cream, mayonnaise, Worcestershire sauce, thyme, about 3/4 cup of the shredded Gruyère or Swiss cheese, and the Parmesan.

Fold the caramelized onions into the cheese mixture. Taste and adjust the salt and pepper carefully, remembering that the cheeses and Worcestershire sauce already add savory flavor.

Transfer the dip to a small 1 1/2- to 2-quart baking dish and spread it into an even layer. Sprinkle the remaining Gruyère or Swiss cheese over the top.

Bake for 20–25 minutes, until the dip is hot and bubbling around the edges. For a more browned top, broil for 1–2 minutes at the end while watching closely.

Let the dip rest for 5–10 minutes before serving so it thickens slightly and is less likely to separate. Serve warm with baguette slices, crackers, potato chips, or cut vegetables.