Preheat the oven to 400°. In a food processor, pulse the olives and garlic until finely chopped, about 10 seconds. Add the cream cheese, goat cheese, mayonnaise, and Parmigiano-Reggiano and process until blended, stopping to scrape down the side of the bowl as necessary. Season the dip with salt and pepper and pulse to blend.
Spread the fig jam in two 10 1/2-by-6-inch ceramic baking dishes. Spoon the goat cheese mixture on top. In a small bowl, toss the panko with the olive oil. Scatter the panko over the cheese and bake for about 20 minutes, until the panko is golden and the dip puffs slightly. Serve hot, with pita chips and crudites.