For pickled onion and avocado, combine onion and avocado in a medium bowl. Pour 1 cup brine from pickle jar over top. Chill, covered, until ready to serve.
Stir together zest, oregano, garlic powder, onion powder, salt, cumin, and pepper in a large bowl. Add ground beef (and/or pork); mix until combined. Working on parchment paper, flatten mixture into a 4x8-inch rectangle, about 1/4-inch thick. Cut into 4 equal rectangular portions to match size of bread.
Heat a griddle or very large nonstick skillet over high heat. Brush hot surface with oil. Working in batches and adding more oil if needed, add beef portions to griddle. Cook, turning halfway through, until edges are crispy and well browned, about 4 minutes. Top each patty with 2 or 3 pickle chips, 1 ham slice, and 1 cheese slice. Cover with a metal bowl, lid, or foil. Cook about 1 minute more, or until cheese is melted and patties are cooked through (160°F).
Meanwhile, generously spoon Golden Aioli on cut sides of bread. Transfer cooked patties to prepared bread bottoms and top with drained pickled onion and avocado, then add bread tops, aioli side down. Place sandwiches on hot griddle; press with a sturdy spatula to smash slightly. Cook, turning halfway through, until bread is toasted, 3 to 4 minutes
