To make the caramelized onions:
Place a large skillet over low heat and when it is hot, add the oil. Add the remaining ingredients and cook, stirring only occasionally as necessary, until the onions are deeply browned and slightly gooey, about 40 minutes. If the onions dry out, add water, 1 tablespoon at a time. Use immediately or cover and refrigerate for up to 2 days.
For the burgers:
Place the chickpeas, onions, panko, garlic, cilantro, tahini, lemon juice and rosemary in a large mixing bowl and mix until well combined. Divide the mixture into 4 portions and form each into a patty about 3⁄4 to 1 inch thick, tossing it back and forth between your hands. Cover and refrigerate for at least 4 hours and up to overnight.
Sprinkle the patties with the salt and pepper. Place a cast iron skillet over high heat and when it is hot but not smoking add the burgers to the dry pan. Cook until well seared on both sides and heated throughout, 8 to 10 minutes. Serve immediately, garnished with lemon quarters.
Beverage pairing: Mastroberardino Greco di Tufo, Italy. All the savory, earthy flavors from the chickpeas, tahini, garlic, and cilantro find resonance with a wine from the south of Italy. Greco di Tufo comes from a hot, dry climate close to the Mediterranean, just like many of the ingredients do. Its rich, savory flavors and smooth, viscous body will animate the combination.